DGGBAL (3lapack)

balance a pair of general real matrices (A,B)

SYNOPSIS

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

      CHARACTER      JOB

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

      DOUBLE         PRECISION A( LDA, * ), B( LDB, * ), LSCALE( * ), RSCALE(
                     * ), WORK( * )

PURPOSE

  DGGBAL 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) 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  (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 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) DOUBLE PRECISION 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.