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

Abstract

In engineering and applied sciences, effectively solving sparse linear systems often requires iterative methods. However, these methods typically involve a high number of iterations. To address this issue and achieve faster convergence, this study introduces a modified approach known as the "third refinement" of the Gauss-Seidel algorithm for solving linear systems. The main aim is to improve convergence speed by reducing both the number of iterations and the spectral radius. This approach involves decomposing the coefficient matrix using a standard splitting strategy and performing interpolation on the resulting simpler matrices. The study investigates and confirms the convergence of this accelerated technique for certain types of matrices. Numerical experiments demonstrate significant improvements in efficiency compared to previous modifications of the algorithm.