Second degree generalized gauss-Seidel iteration method for solving linear system of equations

Tesfaye Kebede

doi:10.4314/ejst.v7i2.5

Abstract

In this paper, a second degree generalized Gauss –Seidel iteration (SDGGS) method for solving linear system of equations whose iterative matrix has real and complex eigenvalues are less than unity in magnitude is presented. Few numerical examples are considered to show the efficiency of the new method compared to first degree Gauss-Seidel (GS), first degree Generalized Gauss-Seidel (GGS) and Second degree Gauss-Seidel (SDGS) methods. It is observed that the spectral radius of the new Second degree Generalized Gauss-Seidel (SDGGS) method is less than the spectral radius of the methods GS, GGS and SDGS. By use of second degree iteration (SD) method, it is possible to accelerate the convergence of any iterative method.Key words: Gauss-Seidel method (GS); Generalized Gauss-Seidel method (GGS); Strictly Diagonally Dominant Matrix.

Highlights

Consider a class of linear stationary second degree methods for solving linear system A x =b (1)where A is a given real non singular n x n matrix and b is a given vector or nx1 matrix (Saad,1995)
It is observed from the table that the same solution is obtained at the 8th iteration by Gauss-Seidel method (GS), at the 6th iteration by Generalized Gauss-Seidel method(GGS) and by Second degree Generalized Gauss-Seidel (SDGGS) method by considering only m=1
For Positive definite (PD) matrix SDGGS method is faster than first degree GS, generalized Gauss-Seidel (GGS) and Second degree GS method

Summary

Introduction

Consider a class of linear stationary second degree methods for solving linear system A x =b (1)where A is a given real non singular n x n matrix and b is a given vector or nx1 (column) matrix (Saad ,1995). A review of Second degree generalized Gauss-Seidel iterative method (SDGGS) is presented. Theorem1.1:- If matrix A is strictly diagonally dominant, the associated generalized Gauss-Seidel iteration method converges for any initial approximation, x (0). The generalized Gauss-Seidel iterative method converges for any initial approximation x(0) .

Results

Conclusion