sgeesx(3F)

NAME
     SGEESX - 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 SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI,
                        VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK,
                        BWORK, INFO )

         CHARACTER      JOBVS, SENSE, SORT

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

         REAL           RCONDE, RCONDV

         LOGICAL        BWORK( * )

         INTEGER        IWORK( * )

         REAL           A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( *
                        )

         LOGICAL        SELECT

         EXTERNAL       SELECT

PURPOSE
     SGEESX 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; 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 real 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 REAL 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 are.  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 may be set to
             N+3 (see INFO below).

     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) REAL array, dimension (LDA, N)
             On entry, the N-by-N matrix A.  On exit, A is 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) REAL array, dimension (N)
             WI      (output) REAL 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 appear consecutively with the eigenvalue having the
             positive imaginary part first.

     VS      (output) REAL 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, 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) REAL 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(1,3*N).  Also, if
             SENSE = 'E' or 'V' or 'B', LWORK >= N+2*SDIM*(N-SDIM), where SDIM
             is the number of selected eigenvalues computed by this routine.
             Note that N+2*SDIM*(N-SDIM) <= N+N*N/2.  For good performance,
             LWORK must generally be larger.

     IWORK   (workspace) INTEGER array, dimension (LIWORK)
             Not referenced if SENSE = 'N' or 'E'.

     LIWORK  (input) INTEGER
             The dimension of the array IWORK.  LIWORK >= 1; if SENSE = 'V' or
             'B', LIWORK >= SDIM*(N-SDIM).

     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 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.