SGEESX(3) LAPACK driver routine (version 3.1) SGEESX(3)
NAME
SGEESX - 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 eigenval-
ues, the real Schur form T, and, optionally, the matrix of Schur vec-
tors Z. This gives the Schur factorization 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 quan-
tities are called s and sep respectively).
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 diago-
nal of the Schur form. = 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).
SELECT (external procedure) 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 ref-
erenced. 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 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 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 subspace 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 eigen-
values appear consecutively with the eigenvalue having the pos-
itive 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 refer-
enced 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 refer-
enced if SENSE = 'N' or 'E'.
WORK (workspace/output) REAL array, dimension (MAX(1,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. Note also
that an error is only returned if LWORK < max(1,3*N), but if
SENSE = 'E' or 'V' or 'B' this may not be large enough. For
good performance, LWORK must generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates upper bounds on the optimal sizes of the arrays
WORK and IWORK, returns these values as the first entries of
the WORK and IWORK arrays, and no error messages related to
LWORK or LIWORK are issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array IWORK. LIWORK >= 1; if SENSE = 'V'
or 'B', LIWORK >= SDIM*(N-SDIM). Note that SDIM*(N-SDIM) <=
N*N/4. Note also that an error is only returned if LIWORK < 1,
but if SENSE = 'V' or 'B' this may not be large enough.
If LIWORK = -1, then a workspace query is assumed; the routine
only calculates upper bounds on the optimal sizes of the arrays
WORK and IWORK, returns these values as the first entries of
the WORK and IWORK arrays, and no error messages related to
LWORK or LIWORK are 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 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 con-
tains the transformation which reduces A to its partially con-
verged 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.
LAPACK driver routine (version 3.N1o)vember 2006 SGEESX(3)