Abstract

This work presents advanced methodologies for the improvement of the Jacobi method by incorporating Gauss-Seidel iterative schemes [1][2]. The Jacobi method, though simple to use [3][4], usually suffers from slow convergence, especially for large-scale linear systems [5]. In this regard, a hybrid scheme is presented that utilizes the strengths of both the Jacobi and Gauss-Seidel methods [6][7]. The important techniques that form the basis of this study include successive over-relaxation (SOR) to accelerate the convergence [2][8], adaptive step-sizing based on residual monitoring [9][10], and preconditioning techniques that enhance the numerical properties of the linear systems [11][12]. Furthermore, the effect of reordering equations for improved convergence rates is also considered [11]. Numerical experiments performed in the context of this study show that these enhancements significantly reduce the number of iterations required for acceptable solutions, thus enhancing computational efficiency [2][13]. In this context, the proposed hybrid scheme outperforms the traditional Jacobi and Gauss-Seidel methods for several scenarios and provides a robust solution for practitioners working in applied mathematics and engineering disciplines [2]. Keywords: Jacobi Method, Gauss-Seidel Method, Iterative Methods, Linear Systems, Convergence Analysis, Successive Over, Relaxation (SOR), Preconditioning.