On the convergence of nonstationary iterative methods for symmetric positive (semi)definite systems

Abstract

We study the linear iterative methods for solving a linear system Ax=b, where A is a symmetric positive definite or singular symmetric positive semidefinite matrix, the system is consistent in case A is singular, i.e., b∈ R(A) , the range of A. We prove that if the stationary iterative methods associated with the nonstationary iterative method satisfy the convergence condition “uniformly”, then this nonstationary iterative method is convergent or, in case the linear system is singular, of quotient convergence. As applications of our convergence results we discuss the convergence of some nonstationary two-stage iterative methods. By using our new results one can complement convergence results of many nonstationary iterative methods and give very simple proofs.