SYNOPSIS
 SUBROUTINE ZGGBAL(
 JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO )
 CHARACTER JOB
 INTEGER IHI, ILO, INFO, LDA, LDB, N
 DOUBLE PRECISION LSCALE( * ), RSCALE( * ), WORK( * )
 COMPLEX*16 A( LDA, * ), B( LDB, * )
PURPOSE
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 1norm 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) 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 (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,N).
 B (input/output) 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 (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,...,ILO1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N.
 LSCALE (output) 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,...,ILO1 = 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 ILO1.
 RSCALE (output) 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,...,ILO1 = 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 ILO1.
 WORK (workspace) REAL 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 (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
FURTHER DETAILS
See R.C. WARD, Balancing the generalized eigenvalue problem,SIAM J. Sci. Stat. Comp. 2 (1981), 141152.