DGEBAL (3lapack)

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 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 matrix, and improve the accuracy of
  the computed eigenvalues and/or eigenvectors.

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 eigenvalues 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.