ZGGBAL balances a pair of general complex 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.

JOB

          JOB is 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

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

A

          A is COMPLEX*16 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

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

B

          B is COMPLEX*16 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

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

ILO

          ILO is INTEGER

IHI

          IHI is 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

          LSCALE is DOUBLE PRECISION 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

          RSCALE is DOUBLE PRECISION 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
            RSCALE(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

          WORK is DOUBLE PRECISION array, dimension (lwork)
          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
          at least 1 when JOB = 'N' or 'P'.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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