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NAME
     zgeesx - compute for an N-by-N complex  nonsymmetric  matrix
     A,  the  eigenvalues, the Schur form T, and, optionally, the
     matrix of Schur vectors Z

SYNOPSIS
     SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE,  N,  A,  LDA,
               SDIM,  W,  VS,  LDVS, RCONDE, RCONDV, WORK, LWORK,
               RWORK, BWORK, INFO )

     CHARACTER JOBVS, SENSE, SORT

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

     DOUBLE PRECISION RCONDE, RCONDV

     LOGICAL BWORK( * )

     DOUBLE PRECISION RWORK( * )

     COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )

     LOGICAL SELECT

     EXTERNAL SELECT



     #include <sunperf.h>

     void zgeesx(char jobvs, char  sort,  int  (*select)(),  char
               sense,  int  n,  doublecomplex  *za,  int lda, int
               *sdim, doublecomplex * w, doublecomplex  *vs,  int
               ldvs, double *rconde, double *rcondv, int *info) ;

PURPOSE
     ZGEESX computes for an N-by-N complex nonsymmetric matrix A,
     the  eigenvalues,  the  Schur  form  T, and, optionally, the
     matrix of Schur vectors Z.  This gives the Schur  factoriza-
     tion A = Z*T*(Z**H).

     Optionally, it also orders the eigenvalues on  the  diagonal
     of  the  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  respec-
     tively).

     A complex matrix is in Schur form if it is upper triangular.


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 one  COMPLEX*16  argu-
               ment
               SELECT must be declared EXTERNAL  in  the  calling
               subroutine.   If  SORT  =  'S',  SELECT is used to
               select eigenvalues to order to the top left of the
               Schur  form.   If SORT = 'N', SELECT is not refer-
               enced.   An  eigenvalue  W(j)   is   selected   if
               SELECT(W(j)) is true.

     SENSE     (input) CHARACTER*1
               Determines which reciprocal condition numbers  are
               computed.  = 'N': None are computed;
               = 'E': Computed for  average  of  selected  eigen-
               values only;
               = 'V': Computed for selected right invariant  sub-
               space 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) COMPLEX*16 array,  dimension  (LDA,
               N)
               On entry, the N-by-N matrix  A.   On  exit,  A  is
               overwritten by its 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 for which SELECT is true.

     W         (output) COMPLEX*16 array, dimension (N)
               W contains the computed eigenvalues, in  the  same
               order that they appear on the diagonal of the out-
               put Schur form T.

     VS        (output) COMPLEX*16 array, dimension (LDVS,N)
               If JOBVS = 'V', VS contains the unitary  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)  COMPLEX*16  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(1,2*N).   Also,  if SENSE = 'E' or 'V' or 'B',
               LWORK >= 2*SDIM*(N-SDIM), where SDIM is the number
               of  selected eigenvalues computed by this routine.
               Note that 2*SDIM*(N-SDIM) <= N*N/2.  For good per-
               formance, LWORK must generally be larger.

     RWORK     (workspace) DOUBLE PRECISION array, dimension (N)

     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 W  con-
               tain  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  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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