Enhancing Linear System Solving Through Third Refinement of Successive and Accelerated Over-Relaxation Methods

Abstract

One of the primary difficulties in linear algebra, considering its widespread application in many different domains, is solving linear system of equations. It is nevertheless apparent that there is a need for a quick, effective approach that can handle a variety of linear systems. In the realm of large and sparse systems, iterative methods play a crucial role in finding solutions. This research paper makes a significant contribution by introducing an enhancement to the current methodology Successive and Accelerated Over Relaxation methods, referred to as the "Third Refinement of Successive and Accelerated Over Relaxation Methods." This new iterative approach demonstrates its effectiveness when applied to coefficient matrices exhibiting properties such as 𝑀- matrix, irreducible diagonal dominance, positive definiteness and symmetry characteristics. Significantly, the proposed method substantially reduces the spectral radius, resulting in fewer iterations and notably enhancing the convergence rate. Numerical experiments were conducted to evaluate its performance compared to existing second refinement of Successive and Accelerated Over Relaxation methods. The outcomes underscore the "Third Refinement of Successive and Accelerated Over Relaxation" methods potentially to boost the efficiency of solving linear systems, thus rendering it a valuable asset within the arsenal of numerical methodologies utilized in scientific and engineering realms.