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NAME
     sgebal - balance a general real matrix A

SYNOPSIS
     SUBROUTINE SGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )

     CHARACTER JOB

     INTEGER IHI, ILO, INFO, LDA, N

     REAL A( LDA, * ), SCALE( * )



     #include <sunperf.h>

     void sgebal(char job, int n, float *sa, int lda,  int  *ilo,
               int *ihi, float *sscale, int *info) ;

PURPOSE
     SGEBAL balances a general real  matrix  A.   This  involves,
     first, permuting A by a similarity transformation to isolate
     eigenvalues in the first 1 to ILO-1 and last IHI+1 to N ele-
     ments on the diagonal; and second, applying a diagonal simi-
     larity transformation to rows and columns ILO to IHI to make
     the  rows  and  columns  as close in norm as possible.  Both
     steps are optional.

     Balancing may reduce the 1-norm of the matrix,  and  improve
     the  accuracy  of  the computed eigenvalues and/or eigenvec-
     tors.


ARGUMENTS
     JOB       (input) CHARACTER*1
               Specifies the operations to be performed on A:
               = 'N':  none:  simply  set  ILO  =  1,  IHI  =  N,
               SCALE(I)  =  1.0  for i = 1,...,N; = 'P':  permute
               only;
               = 'S':  scale only;
               = 'B':  both permute and scale.

     N         (input) INTEGER
               The order of the matrix A.  N >= 0.

     A         (input/output) REAL array, dimension (LDA,N)
               On entry, the input matrix  A.   On  exit,   A  is
               overwritten by the balanced matrix.  If JOB = 'N',
               A is not referenced.  See  Further  Details.   LDA
               (input) INTEGER The leading dimension of the array
               A.  LDA >= max(1,N).

     ILO       (output) INTEGER
               IHI     (output) INTEGER ILO and IHI  are  set  to
               integers such that on exit A(i,j) = 0 if i > j and
               j = 1,...,ILO-1 or I = IHI+1,...,N.  If JOB =  'N'
               or 'S', ILO = 1 and IHI = N.

     SCALE     (output) REAL array, dimension (N)
               Details of the permutations  and  scaling  factors
               applied to A.  If P(j) is the index of the row and
               column interchanged with row and column j and D(j)
               is the scaling factor applied to row and column j,
               then SCALE(j) = P(j)    for j = 1,...,ILO-1 = D(j)
               for j = ILO,...,IHI = P(j)    for j = IHI+1,...,N.
               The order in which the interchanges are made is  N
               to IHI+1, then 1 to ILO-1.

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

FURTHER DETAILS
     The permutations consist  of  row  and  column  interchanges
     which put the matrix in the form

                ( T1   X   Y  )
        P A P = (  0   B   Z  )
                (  0   0   T2 )

     where T1 and T2 are upper triangular matrices  whose  eigen-
     values  lie  along the diagonal.  The column indices ILO and
     IHI mark the starting and ending columns of the submatrix B.
     Balancing   consists   of  applying  a  diagonal  similarity
     transformation inv(D) * B * D to make the  1-norms  of  each
     row  of  B  and  its corresponding column nearly equal.  The
     output matrix is

        ( T1     X*D          Y    )
        (  0  inv(D)*B*D  inv(D)*Z ).
        (  0      0           T2   )

     Information about the permutations P and the diagonal matrix
     D is returned in the vector SCALE.

     This subroutine is based on the EISPACK routine BALANC.

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