Third refinement Generalized Jacobi iterative method for solving linear system of equations

Abstract

The Jacobi and Gauss-Seidel algorithms are among the stationary iterative methods for solving linear system of equations. Obtaining an approximation for the majority of sparse linear systems found in engineering and applied sciences requires efficient iteration approaches. Solving such linear systems using iterative techniques is possible, but the number of iterations is high. To acquire approximate solutions with rapid convergence, the need arises to redesign or make changes to the current approaches. In this study, a modified approach, termed the "third refinement Generalized" of the Jacobi algorithm, for solving linear systems is proposed. The primary objective of this research is to optimize for convergence speed by reducing the number of iterations and the spectral radius. Decomposing the coefficient matrix using a standard splitting strategy and performing an interpolation operation on the resulting simpler matrices led to the development of the proposed method. The study points out that, using the third refinement generalized of Jacobi method, we obtain a solution of a problem with a minimum number of iteration and obtain a greater rate of convergence than other previous methods like Jacobi, refinement Jacobi, refinement generalized Jacobi and second refinement generalized Jacobi.