 ZGEESX(3) computes 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 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 condition 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 columns 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 quantities 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 diagonal of the Schur form. = 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).
SELECT (external procedure) LOGICAL FUNCTION of one COMPLEX*16 argument
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 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 output 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 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 referenced 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 referenced 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 performance, 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 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 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.