SGEESX (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 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.