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zgeesx


 NAME
      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
      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 respec-
      tively).

      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*16 variable
              SELECT must be declared EXTERNAL in the calling sub-
              routine.  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 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 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 refer-
              enced.

      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 recipro-
              cal 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 recipro-
              cal condition number for the selected right invari-
              ant 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 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.