Convergence Analysis for Double Splitting of Matrices
HS Najafi, S. A. Edalatpanah · 2013
Abstract: For single splittings of matrices, there are well-known convergence and comparison theorems. However, there are a few convergence theorems for double splitting. In this paper, we study this class of iterative methods. Furthermore, this paper gives new convergence results for double splitting of matrices. Keywords: Convergence theorem; single splitting; double splitting; iterative methods; spectral radius. 1. Introduction Consider the following linear system A ,x b= (1) where A R∈ n n× , ,b x R∈ n . Large families of iteration methods for solving (1) take the form x M Nx M b i i i+ − −1 1 1 = + =, 0,1, (2)… where, A .= −M N The splitting of the coefficient matrix, where M is nonsingular, is called a single splitting of A [1].There are many iterative methods based on single splitting; see , e.g., [2-10] and the references therein. The double splitting of A was introduced by Woznicki [1] as follows; Splitting the matrix A in the form, A=P-R+S, (3) called the double splitting of A, where P is a nonsingular matrix and we have the following iterative scheme spanned on three successive iterates, x P Rx P Sx P b i