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dgebal

 NAME
      DGEBAL - balance a general real matrix A

 SYNOPSIS
      SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )

          CHARACTER      JOB

          INTEGER        IHI, ILO, INFO, LDA, N

          DOUBLE         PRECISION A( LDA, * ), SCALE( * )

 PURPOSE
      DGEBAL balances a general real matrix A.  This involves,
      first, permuting A by a similarity transformation to isolate
      eigenvalues in the first 1 to ILO-1 and last IHI+1 to N ele-
      ments on the diagonal; and second, applying a diagonal simi-
      larity 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 matrix, 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:
              = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(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 matrix A.  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.  See Further Details.  LDA
              (input) INTEGER The leading dimension of the array
              A.  LDA >= max(1,N).

      ILO     (output) INTEGER
              IHI     (output) INTEGER ILO and IHI are set to
              integers such that on exit A(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.

      SCALE   (output) DOUBLE PRECISION array, dimension (N)
              Details of the permutations and scaling factors

              applied to A.  If P(j) is the index of the row and
              column interchanged with row and column j and D(j)
              is the scaling factor applied to row and column j,
              then SCALE(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.

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

 FURTHER DETAILS
      The permutations consist of row and column interchanges
      which put the matrix in the form

                 ( T1   X   Y  )
         P A P = (  0   B   Z  )
                 (  0   0   T2 )

      where T1 and T2 are upper triangular matrices whose eigen-
      values lie along the diagonal.  The column indices ILO and
      IHI mark the starting and ending columns of the submatrix B.
      Balancing consists of applying a diagonal similarity
      transformation inv(D) * B * D to make the 1-norms of each
      row of B and its corresponding column nearly equal.  The
      output matrix is

         ( T1     X*D          Y    )
         (  0  inv(D)*B*D  inv(D)*Z ).
         (  0      0           T2   )

      Information about the permutations P and the diagonal matrix
      D is returned in the vector SCALE.

      This subroutine is based on the EISPACK routine BALANC.