ZGEESX - compute for an N-by-N complex nonsymmetric matrix
A, the eigenvalues, the Schur form T, and, optionally, the
matrix of Schur vectors Z
SYNOPSIS
SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA,
SDIM, W, VS, LDVS, RCONDE, RCONDV,
WORK, LWORK, RWORK, BWORK, INFO )
CHARACTER JOBVS, SENSE, SORT
INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
DOUBLE PRECISION RCONDE, RCONDV
LOGICAL BWORK( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ),
WORK( * )
LOGICAL SELECT
EXTERNAL SELECT
PURPOSE
ZGEESX computes for an N-by-N complex nonsymmetric matrix
A, the eigenvalues, the Schur form T, and, optionally, the
matrix of Schur vectors Z. This gives the Schur factoriza
tion A = Z*T*(Z**H). Optionally, it also orders the
eigenvalues on the diagonal of the 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 corre
sponding to the selected eigenvalues (RCONDV). The lead
ing columns of Z form an orthonormal basis for this
invariant subspace.
For further explanation of the reciprocal condition num
bers RCONDE and RCONDV, see Section 4.10 of the LAPACK
Users' Guide (where these quantities are called s and sep
respectively).
A complex matrix is in Schur form if it is upper triangu
lar.
ARGUMENTS
JOBVS (input) CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.
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 one COMPLEX*16 argu
ment
SELECT must be declared EXTERNAL in the calling
subroutine. If SORT = 'S', SELECT is used to
select eigenvalues to order to the top left of the
Schur form. If SORT = 'N', SELECT is not refer
enced. An eigenvalue W(j) is selected if
SELECT(W(j)) is true.
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are
computed. = 'N': None are computed;
= 'E': Computed for average of selected eigenval
ues 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) COMPLEX*16 array, dimension (LDA,
N)
On entry, the N-by-N matrix A. On exit, A is
overwritten by its 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 for which SELECT is true.
W (output) COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues, in the same
order that they appear on the diagonal of the out
put Schur form T.
VS (output) COMPLEX*16 array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the unitary matrix Z
of Schur vectors. If JOBVS = 'N', VS is not ref
erenced.
LDVS (input) INTEGER
The leading dimension of the array VS. LDVS >= 1,
and if JOBVS = 'V', LDVS >= N.
If SENSE = 'E' or 'B', RCONDE contains the recip
rocal condition number for the average of the
selected eigenvalues. Not referenced if SENSE =
'N' or 'V'.
RCONDV (output) DOUBLE PRECISION
If SENSE = 'V' or 'B', RCONDV contains the recip
rocal condition number for the selected right
invariant subspace. Not referenced if SENSE = 'N'
or 'E'.
WORK (workspace/output) COMPLEX*16 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(1,2*N). Also, if SENSE = 'E' or 'V' or 'B',
LWORK >= 2*SDIM*(N-SDIM), where SDIM is the number
of selected eigenvalues computed by this routine.
Note that 2*SDIM*(N-SDIM) <= N*N/2. For good per
formance, LWORK must generally be larger.
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
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 W con
tain 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 reordered
because some eigenvalues were too close to sepa
rate (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.
Man(1) output converted with
man2html