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NAME
     sggbal - balance a pair of general real matrices (A,B)

SYNOPSIS
     SUBROUTINE SGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
               RSCALE, WORK, INFO )

     CHARACTER JOB

     INTEGER IHI, ILO, INFO, LDA, LDB, N

     REAL A( LDA, * ), B( LDB, * ), LSCALE( *  ),  RSCALE(  *  ),
               WORK( * )



     #include <sunperf.h>

     void sggbal(char job, int n, float *sa, int lda, float  *sb,
               int  ldb, int *ilo, int *ihi, float *lscale, float
               *rscale, int *info) ;

PURPOSE
     SGGBAL balances a pair of general real matrices (A,B).  This
     involves, first, permuting A and B by similarity transforma-
     tions to isolate eigenvalues in the first 1 to  ILO$-$1  and
     last IHI+1 to N elements on the diagonal; and second, apply-
     ing a diagonal similarity 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 matrices, and improve
     the accuracy of the computed eigenvalues and/or eigenvectors
     in the generalized eigenvalue problem A*x = lambda*B*x.


ARGUMENTS
     JOB       (input) CHARACTER*1
               Specifies the operations to be performed on A  and
               B:
               = 'N':  none:  simply  set  ILO  =  1,  IHI  =  N,
               LSCALE(I)  =  1.0  and  RSCALE(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 matrices A and B.  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.

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

     B         (input/output) REAL array, dimension (LDB,N)
               On entry, the input matrix  B.   On  exit,   B  is
               overwritten by the balanced matrix.  If JOB = 'N',
               B is not referenced.

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

     ILO       (output) INTEGER
               IHI     (output) INTEGER ILO and IHI  are  set  to
               integers such that on exit A(i,j) = 0 and B(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.

     LSCALE    (output) REAL array, dimension (N)
               Details of the permutations  and  scaling  factors
               applied  to  the left side of A and B.  If P(j) is
               the index of the row interchanged with row j,  and
               D(j)  is the scaling factor applied to row j, then
               LSCALE(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.

     RSCALE    (output) REAL array, dimension (N)
               Details of the permutations  and  scaling  factors
               applied  to the right side of A and B.  If P(j) is
               the index of the column interchanged  with  column
               j,  and  D(j)  is  the  scaling  factor applied to
               column  j,  then  LSCALE(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.

     WORK      (workspace) REAL array, dimension (6*N)

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

FURTHER DETAILS
     See R.C. WARD, Balancing the generalized eigenvalue problem,
                    SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

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