CGEESX (3lapack)

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.