CXML
        	    
LAPACK version 3.0
dgeesx(3)

PURPOSE
  DGEESX - 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

SYNTAX
  SUBROUTINE DGEESX( 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

      DOUBLE	     PRECISION RCONDE, RCONDV

      LOGICAL	     BWORK( * )

      INTEGER	     IWORK( * )

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

      LOGICAL	     SELECT

      EXTERNAL	     SELECT

DESCRIPTION
  DGEESX 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 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 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) DOUBLE PRECISION 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) 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
	  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, and if JOBVS =
	  'V', LDVS >= N.

  RCONDE  (output) DOUBLE PRECISION
	  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) DOUBLE PRECISION
	  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) DOUBLE PRECISION 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/output) INTEGER array, dimension (LIWORK)
	  Not referenced if SENSE = 'N' or 'E'.	 On exit, if INFO = 0,
	  IWORK(1) returns the optimal LIWORK.

  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.