AN EXTENDED PARAMETERIZED ACCELERATED OVER-RELAXATION ITERATIVE METHOD WITH REFINEMENT FOR LINEAR SYSTEMS

Abstract

Most largely sparse linear systems encountered in applied sciences and engineering requires efficient iterative methods to obtain an approximate solution. In this thesis, an Extended Accelerated Over-relaxation (EAOR) iterative method for solving linear systems was developed by introducing a new acceleration parameter to improve convergence rate of family of Accelerated Over-relaxation (AOR) methods. The method was developed through decomposition of the coefficient matrix by the usual splitting approach and interpolation procedure on the sub-matrices. Analysis of convergence of the method were examined on some special matrices like , , and Irreducible weak diagonally dominant matrices. In addition, the Refinement of Extended Accelerated Over-relaxation (REAOR) iterative method was also developed coupled with its convergence properties for the special matrices. To validate the effectiveness of the proposed methods, some numerical tests including problems from fuzzy linear systems and heat transfer were conducted to verify the theoretical results. The results obtained indicated that the spectral radii of the proposed methods are smaller than the compared methods reviewed in the work. Based on the spectral radii results and the convergence results produced, it was concluded that the developed methods converge with lower number of iterations and computational time than the compared methods. This reveals that the introduction of a parameter to the general two-parameter AOR family methods has improved the convergence results. From the results obtained, the EAOR iterative method converges approximately 1.4 times faster than the KAOR iterative method and 1.8 times faster than the Quasi Accelerated Over-Relaxation (QAOR) iterative method. While the REAOR method converges 1.2 times faster than the Refinement of Accelerated Over-relaxation (RAOR) iterative method.