Refinements of Generalized Jacobi Method: A Higher-Order Approach

This study presents a comparative analysis of the pth Refinement of Generalized Jacobi method, focusing on its derivation, convergence properties, and numerical performance. We first derive the Third Refinement of Generalized Jacobi method and establish a general formula applicable to any pth Refinement. The analysis rigorously proves convergence for various types of matrices, including Strictly Diagonally Dominant (SDD) and M-matrices. Additionally, we demonstrate that the convergence rate of the (p + 1)th Refinement surpasses that of the pth Refinement, assuming the Generalized Jacobi method converges. Numerical examples are provided to support our theoretical findings, showcasing the improved convergence rates of the (p + 1)th Refinement compared to the pth Refinement. These results underline the advantages of higher-order refinements in the Generalized Jacobi method.