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cggbal

 NAME
      CGGBAL - balance a pair of general complex matrices (A,B)
      for the generalized eigenvalue problem A*X = lambda*B*X

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

          CHARACTER      JOB

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

          REAL           LSCALE( * ), RSCALE( * ), WORK( * )

          COMPLEX        A( LDA, * ), B( LDB, * )

 PURPOSE
      CGGBAL balances a pair of general complex matrices (A,B) for
      the generalized eigenvalue problem A*X = lambda*B*X.  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 eigenvec-
      tors.

 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 matrices A and B.  N >= 0.

      A       (input/output) COMPLEX array, dimension (LDA,N)
              On entry, the input matrix A.  On exit, A is
              overwritten by the balanced matrix.

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

      B       (input/output) COMPLEX array, dimension (LDB,N)
              On entry, the input matrix B.  On exit, B is
              overwritten by the balanced matrix.

      LDB     (input) INTEGER
              The leading dimension of the matrix 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 row interchanged with row j, and
              D(j) is the scaling factor applied to row 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    (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.