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dgeesx.f(3)				      LAPACK				      dgeesx.f(3)

NAME
       dgeesx.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dgeesx (JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI, VS, LDVS, RCONDE,
	   RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
	    DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
	   vectors for GE matrices

Function/Subroutine Documentation
   subroutine dgeesx (characterJOBVS, characterSORT, logical, externalSELECT, characterSENSE,
       integerN, double precision, dimension( lda, * )A, integerLDA, integerSDIM, double
       precision, dimension( * )WR, double precision, dimension( * )WI, double precision,
       dimension( ldvs, * )VS, integerLDVS, double precisionRCONDE, double precisionRCONDV,
       double precision, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK,
       integerLIWORK, logical, dimension( * )BWORK, integerINFO)
	DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

	    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).

       Parameters:
	   JOBVS

		     JOBVS is CHARACTER*1
		     = 'N': Schur vectors are not computed;
		     = 'V': Schur vectors are computed.

	   SORT

		     SORT is 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

		     SELECT is procedure) 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

		     SENSE is 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

		     N is INTEGER
		     The order of the matrix A. N >= 0.

	   A

		     A is 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

		     LDA is INTEGER
		     The leading dimension of the array A.  LDA >= max(1,N).

	   SDIM

		     SDIM is 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

		     WR is DOUBLE PRECISION array, dimension (N)

	   WI

		     WI is 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

		     VS is 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

		     LDVS is INTEGER
		     The leading dimension of the array VS.  LDVS >= 1, and if
		     JOBVS = 'V', LDVS >= N.

	   RCONDE

		     RCONDE is 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

		     RCONDV is 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

		     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
		     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

	   LWORK

		     LWORK is 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. Note also that an error is only
		     returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
		     'B' this may not be large enough.
		     For good performance, LWORK must generally be larger.

		     If LWORK = -1, then a workspace query is assumed; the routine
		     only calculates upper bounds on the optimal sizes of the
		     arrays WORK and IWORK, returns these values as the first
		     entries of the WORK and IWORK arrays, and no error messages
		     related to LWORK or LIWORK are issued by XERBLA.

	   IWORK

		     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
		     On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

	   LIWORK

		     LIWORK is INTEGER
		     The dimension of the array IWORK.
		     LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
		     Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
		     only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
		     may not be large enough.

		     If LIWORK = -1, then a workspace query is assumed; the
		     routine only calculates upper bounds on the optimal sizes of
		     the arrays WORK and IWORK, returns these values as the first
		     entries of the WORK and IWORK arrays, and no error messages
		     related to LWORK or LIWORK are issued by XERBLA.

	   BWORK

		     BWORK is LOGICAL array, dimension (N)
		     Not referenced if SORT = 'N'.

	   INFO

		     INFO is 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.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Definition at line 280 of file dgeesx.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dgeesx.f(3)
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