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RedHat 9 (Linux i386) - man page for dgeesx (redhat section l)

DGEESX(l)					)					DGEESX(l)

NAME
       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

SYNOPSIS
       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

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  condi-
       tion  number for the average of the selected eigenvalues (RCONDE); and computes a recipro-
       cal condition number for the right invariant subspace corresponding to the selected eigen-
       values  (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 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 com-
	       puted;
	       = '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 sort-
	       ing) 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 ei-
	       genvalues 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.

LAPACK version 3.0			   15 June 2000 				DGEESX(l)


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