ZGEES (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 ZGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS, LDVS, WORK,
                    LWORK, RWORK, BWORK, INFO )

      CHARACTER     JOBVS, SORT

      INTEGER       INFO, LDA, LDVS, LWORK, N, SDIM

      LOGICAL       BWORK( * )

      DOUBLE        PRECISION RWORK( * )

      COMPLEX*16    A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )

      LOGICAL       SELECT

      EXTERNAL      SELECT

PURPOSE

  ZGEES 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.  The leading columns
  of Z then form an orthonormal basis for the invariant subspace
  corresponding to the selected eigenvalues.

  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 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.  The eigenvalue W(j) is selected if SELECT(W(j)) is
          true.

  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 has been 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; if JOBVS = 'V',
          LDVS >= N.

  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).  For good
          performance, 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
          matrix 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.