CGGBAL 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 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 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 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

          RSCALE is 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


            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 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

          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.

November 2011

  See R.C. WARD, Balancing the generalized eigenvalue problem,


                 SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.