SECOND REFINEMENT OF PRECONDITIONED ACCELERATED OVERRELAXATION METHOD FOR SOLUTION OF LINEAR ALGEBRAIC SYSTEM

Abstract

This present work concerns the numerical solution of linear system of algebraic equation 𝐴𝑥 = 𝑏 by second refinement of accelerated overrelaxation (AOR) method. This technique is especially useful in solving linear system arising from discretisation of ordinary differential equations or partial differential equation where the coefficient matrix is an irreducibly diagonally dominant 𝐿- matrix. A suitable preconditioner is applied to the linear system before a second refinement algorithm is processed. As in all iterative methods for linear systems, this is aimed at minimizing the spectral radius in order to reduce the number of iterations needed for convergence. Hence, the SRPAOR method converges faster than AOR, PAOR, RAOR and RPAOR by a factor of 5.75, 3, 2.87 and 1.5 respectively. Optimum convergence is attained when 𝑟 = 1.0, 𝜔 = 1.1 and when 𝑟 = 0.99, 𝜔 = 1.0. Numerical examples proved the efficiency of second refinement of preconditioned AOR over the AOR, preconditioned AOR and first refinement of AOR methods. The techniques of preconditioning and second refinement have been exploited to introduce a new approach towards improving the rate of convergence of the AOR iterative method in solving linear system of equations. The implication of the method indicates an enhancement or modification to the original PAOR method, which led to improved accuracy in solving linear algebraic systems.