zgebal(3F)

NAME
     ZGEBAL - balance a general complex matrix A

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

         CHARACTER      JOB

         INTEGER        IHI, ILO, INFO, LDA, N

         DOUBLE         PRECISION SCALE( * )

         COMPLEX*16     A( LDA, * )

PURPOSE
     ZGEBAL balances a general complex 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) COMPLEX*16 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 CBAL.