An Accelerated Iterative Technique: Third Refinement of Gauss-Seidel Algorithm for Linear Systems

Abstract

This study presents a novel accelerated iterative method referred to as the Third Refinement of the Gauss-Seidel Algorithm (TRGS) for solving large-scale linear systems of equations. By integrating a three-level refinement strategy into the classical Gauss-Seidel method, the proposed technique significantly improves the convergence rate and computational efficiency. The method is rigorously analyzed for consistency, stability, and convergence, and is evaluated through numerical experiments on various benchmark problems. Results demonstrate that the TRGS algorithm outperforms both the traditional Gauss-Seidel and other refinement-based methods in terms of iteration count and solution accuracy. This advancement offers a valuable contribution to numerical linear algebra, particularly in scientific computing where fast and accurate solutions to linear systems are critical.