DGEES (3lapack)

compute for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z

SYNOPSIS

  SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS, LDVS,
                    WORK, LWORK, BWORK, INFO )

      CHARACTER     JOBVS, SORT

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

      LOGICAL       BWORK( * )

      DOUBLE        PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
                    WR( * )

      LOGICAL       SELECT

      EXTERNAL      SELECT

PURPOSE

  DGEES computes for an N-by-N real nonsymmetric matrix A, the eigenvalues,
  the real Schur form T, and, optionally, the matrix of Schur vectors Z.
  This gives the Schur factorization A = Z*T*(Z**T).

  Optionally, it also orders the eigenvalues on the diagonal of the real
  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 matrix is in real Schur form if it is upper quasi-triangular with 1-by-1
  and 2-by-2 blocks. 2-by-2 blocks will be standardized in the form
          [  a  b  ]
          [  c  a  ]

  where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

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 two DOUBLE PRECISION arguments
          SELECT must be declared EXTERNAL in the calling subroutine.  If
          SORT = 'S', SELECT is used to select eigenvalues to sort to the top
          left of the Schur form.  If SORT = 'N', SELECT is not referenced.
          An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
          SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
          conjugate pair of eigenvalues is selected, then both complex
          eigenvalues are selected.  Note that a selected complex eigenvalue
          may no longer satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering,
          since ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned); in this case
          INFO is set to N+2 (see INFO below).

  N       (input) INTEGER
          The order of the matrix A. N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N matrix A.  On exit, A has been overwritten by
          its real 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 (after sorting) for which SELECT is true. (Complex
          conjugate pairs for which SELECT is true for either eigenvalue
          count as 2.)

  WR      (output) DOUBLE PRECISION array, dimension (N)
          WI      (output) DOUBLE PRECISION array, dimension (N) WR and WI
          contain the real and imaginary parts, respectively, of the computed
          eigenvalues in the same order that they appear on the diagonal of
          the output Schur form T.  Complex conjugate pairs of eigenvalues
          will appear consecutively with the eigenvalue having the positive
          imaginary part first.

  VS      (output) DOUBLE PRECISION array, dimension (LDVS,N)
          If JOBVS = 'V', VS contains the orthogonal 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) DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,3*N).  For good
          performance, LWORK must generally be larger.

  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 WR and WI 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.