  Aitken extrapolation normally applied to convergent fixed point iteration is
extended to extrapolate the solution of a divergent iteration. In addition,
higher order Aitken extrapolation is introduced that enables successive
decomposition of high Eigen values of the iteration matrix to enable
convergence. While extrapolation of a convergent fixed point iteration using a
geometric series sum is a known form of Aitken acceleration, it is shown in
this paper that the same formula can be used to estimate the solution of sets
of linear equations from diverging Gauss Seidel iterations. In both convergent
and divergent iterations, the ratios of differences among the consecutive
values of iteration eventually form a convergent or divergent series with a
factor equal to the largest Eigen value of the iteration matrix. Higher order
Aitken extrapolation is shown to eliminate the influence of dominant Eigen
values of the iteration matrix in successive order until the iteration is
determined by the lowest possible Eigen value. For the convergent part of the
Gauss Seidel iteration, further acceleration is made possible by coupling of
the extrapolation technique with the successive over relaxation method.
Application examples from both convergent and divergent iterations have been
provided. Coupling of the extrapolation with the successive over relaxation
technique is also illustrated for a steady state two dimensional heat flow
problem which was solved using MATLAB programming.
