DGEBAK  -  form  the  right or left eigenvectors of a real
       general matrix by backward transformation on the  computed
       eigenvectors of the balanced matrix output by DGEBAL


SYNOPSIS

       SUBROUTINE DGEBAK( JOB,  SIDE,  N,  ILO, IHI, SCALE, M, V,
                          LDV, INFO )

           CHARACTER      JOB, SIDE

           INTEGER        IHI, ILO, INFO, LDV, M, N

           DOUBLE         PRECISION SCALE( * ), V( LDV, * )


PURPOSE

       DGEBAK forms the right or left eigenvectors of a real gen­
       eral  matrix  by  backward  transformation on the computed
       eigenvectors of the balanced matrix output by DGEBAL.


ARGUMENTS

       JOB     (input) CHARACTER*1
               Specifies  the  type  of  backward  transformation
               required: = 'N', do nothing, return immediately; =
               'P', do backward  transformation  for  permutation
               only;  = 'S', do backward transformation for scal­
               ing only; = 'B', do backward  transformations  for
               both  permutation  and  scaling.   JOB must be the
               same as the argument JOB supplied to DGEBAL.

       SIDE    (input) CHARACTER*1
               = 'R':  V contains right eigenvectors;
               = 'L':  V contains left eigenvectors.

       N       (input) INTEGER
               The number of rows of the matrix V.  N >= 0.

       ILO     (input) INTEGER
               IHI     (input) INTEGER The integers ILO  and  IHI
               determined  by DGEBAL.  1 <= ILO <= IHI <= N, if N
               > 0; ILO=1 and IHI=0, if N=0.

       SCALE   (input) DOUBLE PRECISION array, dimension (N)
               Details of the permutation and scaling factors, as
               returned by DGEBAL.

       M       (input) INTEGER
               The number of columns of the matrix V.  M >= 0.

       V       (input/output) DOUBLE PRECISION array, dimension
               (LDV,M)
               On entry, the matrix of right or left eigenvectors
               to  be  transformed,  as  returned  by  DHSEIN  or
               DTREVC.   On  exit,  V  is  overwritten   by   the


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

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


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