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sgeesx


 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 factoriza-
      tion 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 respec-
      tively).

      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 variables
              SELECT must be declared EXTERNAL in the calling sub-
              routine.  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 eigen-
              value 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 sub-
              space 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 posi-
              tive 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 refer-
              enced.

      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 recipro-
              cal 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 recipro-
              cal condition number for the selected right invari-
              ant 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.