The strategies of improvement the Performance for the Jacobi method using Gaussian iterative methods (Comprehensive Journal of Science, 2026)
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.