DGEES(l) ) DGEES(l)
NAME
DGEES - compute for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vec-
tors Z
SYNOPSIS
SUBROUTINE DGEES( 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( * )
DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( * )
LOGICAL SELECT
EXTERNAL SELECT
PURPOSE
DGEES computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, 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. The leading columns of Z then form an orthonormal 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 DOUBLE PRECISION 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 eigen-
values 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N) WR and WI contain the real and imaginary parts, respectively, of the com-
puted eigenvalues in the same order that they appear on the diagonal of the output Schur form T. Complex conjugate pairs of eigen-
values will appear consecutively with the eigenvalue having the positive imaginary part first.
VS (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 ei-
genvalues 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 version 3.0 15 June 2000 DGEES(l)