Convergence Analysis Of The Generalized Alternating Two-Stage Method And Gaoor Iterative Method

Solutions of large-scale sparse linear algebraic systems are deeply involved in various scientific and engineering fields, such as computational electromagnetics, numerical solutions of high-order differential equations, optimization problems, fluid mechanics and reservoir modeling. Moreover, research of methods for solving large-scale sparse systems of linear algebraic equations becomes one of the key issues of large-scale scientific and engineering computing and such research has important theoretic significance and practical applications. In this paper, we study iteration methods to iterative solutions of large-scale sparse linear algebraic equations.Firstly, we give the generalized alternating two-stage method , then we study the convergence of the generalized alternating two-stage method when the coefficient matrix is a monotone matrix, a nonsingular H matrix, a symmetric positive definite matrix, respectively. Secondly, we study the convergence of the generalized alternating two-stage method when the coefficient matrix is an M - matrix with property c , a symmetric positive semi-definite matrix, respectively.Finally, we study the convergence of the GAOR method for solving the linear system Hy = ffor strictly doubly diagonally dominant andα-strictly doubly diagonally dominant coefficient matrices. We obtain bounds for the spectral radius of the matrix L_ω,r which is the iterative matrix of the GAOR iterative method. Moreover, we present convergence theorems of the GAOR method and several numerical examples.