Refinements of Some Iterative Methods for Solving Linear System of Equations

Abstract

The efficient and accurate solution of linear systems of equations is a fundamental problem in various scientific and engineering fields. In this study, we focus on the refinements of iterative methods for solving linear systems of equations (▁A k=▁b). The research proposes two methods namely, third refinement of Jacobi method (TRJ) and third refinement of Gauss-Seidel (TRGS) method, which minimizes the spectral radius of the iteration matrix significantly when compared to any of the initial refinements of Jacobi and Gauss-Seidel methods. The study explores ways to optimize their convergence behavior by incorporating refinement techniques and adaptive strategies. These refinements exploit the structural properties of the coefficient matrix to achieve faster convergence and improved solution accuracy. To evaluate the effectiveness of the proposed refinements, numerical examples were tested to see the efficiency of the proposed TRJ and TRGS on a diverse set of linear equations. We compare the convergence behavior, computational efficiency, and solution accuracy of the refined iterative methods against their traditional counterparts. The experimental results demonstrate significant improvements in terms of convergence rate and computational efficiency when compared to their initial refinements. The proposed refinements have the potential to contribute to the development of more efficient and reliable solvers for linear systems, benefiting various scientific and engineering applications.