SGEES(l) ) SGEES(l)
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
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( * )
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 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. The leading columns of Z
then form an orthonormal basis for the invariant subspace corresponding to the selected
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).
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 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 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 (espe-
cially if the eigenvalue is ill-conditioned); in this case INFO is set to N+2 (see
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
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 sort-
ing) 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 imagi-
nary 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 ei-
genvalues will appear consecutively with the eigenvalue having the positive imagi-
nary 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; 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 generally 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 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-con-
ditioned); = N+2: after reordering, roundoff changed values of some complex eigen-
values so that leading eigenvalues in the Schur form no longer satisfy
SELECT=.TRUE. This could also be caused by underflow due to scaling.
LAPACK version 3.0 15 June 2000 SGEES(l)