Open AccessSecond-refinement of Gauss-Seidel iterative method for solving linear system of equations
Author(s) -
Tesfaye Kebede Enyew,
Gurju Awgichew,
Eshetu Haile,
Gashaye Dessalew Abie
Publication year - 2020
Publication title -
the ethiopian journal of science and technology/ethiopian journal of science and technology
Language(s) - English
Resource type - Journals
eISSN - 2312-6019
pISSN - 1816-3378
DOI - 10.4314/ejst.v13i1.1
Subject(s) - gauss–seidel method , mathematics , spectral radius , iterative method , coefficient matrix , linear system , convergence (economics) , rate of convergence , diagonally dominant matrix , matrix (chemical analysis) , gauss , system of linear equations , positive definite matrix , mathematical analysis , mathematical optimization , pure mathematics , computer science , eigenvalues and eigenvectors , key (lock) , physics , computer security , materials science , composite material , quantum mechanics , economics , invertible matrix , economic growth
Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.