Abstract
In this paper, Aitken’s 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 that in this paper, 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 (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 values. 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 (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.
Highlights
Iterative solutions to systems of equations are widely employed for solving scientific problems
Chebyshev acceleration is a way of transforming the iteration sequence which, for iteration matrix of known upper and lower bound Eigen values, the transformed sequence using Chebyshev polynomials leads to convergence of the fixed point iteration
For solving a system of linear equations using fixed point iteration, the Aitken extrapolation formula can be written in the form: x ek k where: x = The estimate of the solution at the limit of iteration; ek = The difference in consecutive x values, i.e., xk 1 xk ;
Summary
Iterative solutions to systems of equations are widely employed for solving scientific problems. The geometric series sum based form of Aitken extrapolation for fixed point iteration involving the system of equations AX B is based on the formula [6]: X. where X is the Aitken extrapolation of the solution, Xk is the approximation to the solution at the kth iteration, Ek is the difference in X at the kth iteration X k 1 – X k and JAMP. Chebyshev acceleration is a way of transforming the iteration sequence which, for iteration matrix of known upper and lower bound Eigen values, the transformed sequence using Chebyshev polynomials leads to convergence of the fixed point iteration. Thereafter, application examples of both convergent and divergent fixed point iterations follow that include finite difference solution of Laplace equation to a two dimensional heat flow problem which was solved using MATLAB programming
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