Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations

Newton-HSS methods, that are variants of inexact Newton methods different from Newton-Krylov methods, have been shown to be competitive methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices [Bai and Guo, 2010]. In that paper, only local convergence was proved. In this paper, we prove a Kantorovich-type semilocal convergence. Then we introduce Newton-HSS methods with a backtracking strategy and analyse their global convergence. Finally, these globally convergent Newton-HSS methods are shown to work well on several typical examples using different forcing terms to stop the inner iterations.