SGEEV  - compute for an N-by-N real nonsymmetric matrix A,
       the eigenvalues and, optionally,  the  left  and/or  right
       eigenvectors


SYNOPSIS

       SUBROUTINE SGEEV( JOBVL,  JOBVR,  N,  A,  LDA, WR, WI, VL,
                         LDVL, VR, LDVR, WORK, LWORK, INFO )

           CHARACTER     JOBVL, JOBVR

           INTEGER       INFO, LDA, LDVL, LDVR, LWORK, N

           REAL          A( LDA, * ), VL( LDVL, * ), VR( LDVR,  *
                         ), WI( * ), WORK( * ), WR( * )


PURPOSE

       SGEEV  computes  for an N-by-N real nonsymmetric matrix A,
       the eigenvalues and, optionally,  the  left  and/or  right
       eigenvectors.  The right eigenvector v(j) of A satisfies
                        A * v(j) = lambda(j) * v(j)
       where lambda(j) is its eigenvalue.
       The left eigenvector u(j) of A satisfies
                     u(j)**H * A = lambda(j) * u(j)**H
       where u(j)**H denotes the conjugate transpose of u(j).

       The computed eigenvectors are normalized to have Euclidean
       norm equal to 1 and largest component real.


ARGUMENTS

       JOBVL   (input) CHARACTER*1
               = 'N': left eigenvectors of A are not computed;
               = 'V': left eigenvectors of A are computed.

       JOBVR   (input) CHARACTER*1
               = 'N': right eigenvectors of A are not computed;
               = 'V': right eigenvectors of A are computed.

       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.

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

       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.   Complex

               tively with the  eigenvalue  having  the  positive
               imaginary part first.

       VL      (output) REAL array, dimension (LDVL,N)
               If  JOBVL  =  'V',  the left eigenvectors u(j) are
               stored one after another in the columns of VL,  in
               the  same  order as their eigenvalues.  If JOBVL =
               'N', VL is not referenced.  If the j-th eigenvalue
               is  real,  then u(j) = VL(:,j), the j-th column of
               VL.  If the j-th and (j+1)-st eigenvalues  form  a
               complex  conjugate  pair,  then  u(j)  = VL(:,j) +
               i*VL(:,j+1) and
               u(j+1) = VL(:,j) - i*VL(:,j+1).

       LDVL    (input) INTEGER
               The leading dimension of the array VL.  LDVL >= 1;
               if JOBVL = 'V', LDVL >= N.

       VR      (output) REAL array, dimension (LDVR,N)
               If  JOBVR  =  'V', the right eigenvectors v(j) are
               stored one after another in the columns of VR,  in
               the  same  order as their eigenvalues.  If JOBVR =
               'N', VR is not referenced.  If the j-th eigenvalue
               is  real,  then v(j) = VR(:,j), the j-th column of
               VR.  If the j-th and (j+1)-st eigenvalues  form  a
               complex  conjugate  pair,  then  v(j)  = VR(:,j) +
               i*VR(:,j+1) and
               v(j+1) = VR(:,j) - i*VR(:,j+1).

       LDVR    (input) INTEGER
               The leading dimension of the array VR.  LDVR >= 1;
               if JOBVR = 'V', LDVR >= N.

       WORK    (workspace/output) REAL 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),  and  if  JOBVL  = 'V' or JOBVR = 'V',
               LWORK >= 4*N.  For good  performance,  LWORK  must
               generally be larger.

               If  LWORK = -1, then a workspace query is assumed;
               the routine only calculates the  optimal  size  of
               the  WORK  array,  returns this value as the first
               entry of the WORK  array,  and  no  error  message
               related to LWORK is issued by XERBLA.

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value.

               pute all the eigenvalues, and no eigenvectors have
               been computed; elements i+1:N of WR and WI contain
               eigenvalues which have converged.


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