cgeesx(3F)

NAME
     CGEESX - 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 CGEESX( 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

         REAL           RCONDE, RCONDV

         LOGICAL        BWORK( * )

         REAL           RWORK( * )

         COMPLEX        A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )

         LOGICAL        SELECT

         EXTERNAL       SELECT

PURPOSE
     CGEESX 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  (input) LOGICAL FUNCTION of one COMPLEX 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 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 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 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) REAL
             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) REAL
             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 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 performance, LWORK
             must generally be larger.

     RWORK   (workspace) REAL 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.