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DGEESX(1)	      LAPACK driver routine (version 3.2)	     DGEESX(1)

NAME
       DGEESX  - computes for an N-by-N real nonsymmetric matrix A, the eigen‐
       values, 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 eigenval‐
       ues,  the  real Schur form T, and, optionally, the matrix of Schur vec‐
       tors Z.	This gives the Schur factorization A  =	 Z*T*(Z**T).   Option‐
       ally,  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	 quan‐
       tities 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 diago‐
	       nal of the Schur form.  = 'N': Eigenvalues are not ordered;
	       = 'S': Eigenvalues are ordered (see SELECT).

       SELECT  (external 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 ref‐
	       erenced.	 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 ei‐
	       genvalue 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  ei‐
	       genvalues  (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 refer‐
	       enced 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 refer‐
	       enced if SENSE = 'N' or 'E'.

       WORK	 (workspace/output)   DOUBLE   PRECISION   array,    dimension
       (MAX(1,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.  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	calcu‐
	       lates  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   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
	       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).  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   (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 con‐
	       tains the transformation which reduces A to its partially  con‐
	       verged  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 driver routine (version 3November 2008			     DGEESX(1)
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