Abstract
This paper presents generalized refinement of Gauss-Seidel method of solving system of linear equations by considering consistently ordered 2-cyclic matrices. Consistently ordered 2-cyclic matrices are obtained while finite difference method is applied to solve differential equation. Suitable theorems are introduced to verify the convergence of this proposed method. To observe the effectiveness of this method, few numerical examples are given. The study points out that, using the generalized refinement of Gauss-Seidel 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.
Highlights
Consider the problem of large and sparse linear systems of the formAx = b, ð1Þ where A = ðaijÞ is a nonsingular real matrix of order n, b is a given n-dimensional real vector, and x is an n-dimensional vector to be determined
In this paper, we study the generalized refinement of Gauss-Seidel (GRGS) iterative method which is used to accelerate the convergence of the basic Gauss-Seidel method
Large and sparse linear systems of equations which arise from the discretization of PDE and ODE problems are often solved by iterative methods
Summary
Research results show that generalized, refinement, and extrapolation (acceleration or relaxation) are used for modifying the Gauss-Seidel method. Salkuyeh [1] introduced the generalized Gauss-Seidel (GGS) method and discussed the convergence of the method by considering strictly diagonally dominant (SDD) and M-matrices. The refinement of the Gauss-Seidel (RGS) method was studied by Vatti and Eneyew [2] and proved the convergence of the method by taking SDD matrices. Enyew et al [3] developed a second refinement of the Gauss-Seidel method on SDD, symmetric positive definite (SPD), and M-matrices. In this paper, we study the generalized refinement of Gauss-Seidel (GRGS) iterative method which is used to accelerate the convergence of the basic Gauss-Seidel method.
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