dgees(3F)

NAME
     DGEES - 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.