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NAME
     sgees - 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 SGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM,  WR,
               WI, VS, LDVS, WORK, LWORK, BWORK, INFO )

     CHARACTER JOBVS, SORT

     INTEGER INFO, LDA, LDVS, LWORK, N, SDIM

     LOGICAL BWORK( * )

     REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( * )

     LOGICAL SELECT

     EXTERNAL SELECT



     #include <sunperf.h>

     void sgees(char jobvs, char sort, int  (*select)(),  int  n,
               float  *sa,  int  lda, int *sdim, float *wr, float
               *wi, float *vs, int ldvs, int *info) ;

PURPOSE
     SGEES 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  factoriza-
     tion 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.  The leading columns of Z then form an  ortho-
     normal basis for the invariant subspace corresponding to the
     selected eigenvalues.

     A 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': Eigen-
               values 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 refer-
               enced.   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  eigen-
               values  is selected, then both complex eigenvalues
               are selected.  Note that a selected complex eigen-
               value  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 is set to N+2 (see INFO below).

     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  has
               been 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, respec-
               tively, of the computed eigenvalues  in  the  same
               order that they appear on the diagonal of the out-
               put Schur form  T.   Complex  conjugate  pairs  of
               eigenvalues  will  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;
               if JOBVS = 'V', LDVS >= N.

     WORK      (workspace/output) REAL array, dimension (LWORK)
               On exit, if INFO = 0, WORK(1) contains the optimal
               LWORK.

     LWORK     (input) INTEGER
               The  dimension  of  the  array  WORK.   LWORK   >=
               max(1,3*N).  For good performance, LWORK must gen-
               erally be larger.

     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  ille-
               gal 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  matrix  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  eigen-
               values   in  the  Schur  form  no  longer  satisfy
               SELECT=.TRUE.  This could also be caused by under-
               flow due to scaling.

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