ZGEESX(1) LAPACK driver routine (version 3.2) ZGEESX(1)NAME
ZGEESX - computes for an N-by-N complex nonsymmetric matrix A, the ei‐
genvalues, the Schur form T, and, optionally, the matrix of Schur vec‐
tors 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 eigen‐
values, the Schur form T, and, optionally, the matrix of Schur vectors
Z. This gives the Schur factorization 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 condi‐
tion 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 col‐
umns 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 complex matrix is in Schur form if it is upper triangular.
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 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
referenced. 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 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) 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 ei‐
genvalues 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 output Schur form T.
VS (output) COMPLEX*16 array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the unitary matrix Z of Schur vec‐
tors. 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) DOUBLE PRECISION
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) DOUBLE PRECISION
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) COMPLEX*16 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(1,2*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. Note also that an
error is only returned if LWORK < max(1,2*N), but if SENSE =
'E' or 'V' or 'B' this may not be large enough. For good per‐
formance, LWORK must generally be larger. If LWORK = -1, then
a workspace query is assumed; the routine only calculates upper
bound on the optimal size of the array WORK, returns this value
as the first entry of the WORK array, and no error message
related to LWORK is issued by XERBLA.
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 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 W contain those ei‐
genvalues 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 separate (the problem 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 driver routine (version 3November 2008 ZGEESX(1)
