sggbal(3F)

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( * )

PURPOSE
     SGGBAL balances a pair of general real matrices (A,B).  This involves,
     first, permuting A and B by similarity transformations to isolate
     eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N elements on the
     diagonal; and second, applying 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 illegal value.

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