DGEES(l) ) DGEES(l)NAME
DGEES - compute 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
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 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. The leading columns
of Z then form an orthonormal basis for the invariant subspace corre‐
sponding 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 diago‐
nal 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 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 complex
eigenvalues are selected. 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 eigen‐
values (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 ei‐
genvalues (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
computed eigenvalues in the same order that they appear on the
diagonal of the output Schur form T. Complex conjugate pairs
of eigenvalues 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 con‐
tains 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 prob‐
lem is very ill-conditioned); = N+2: after reordering, roundoff
changed values of some complex eigenvalues so that leading ei‐
genvalues 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)
