Multi-step modified Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices

Abstract

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iterative method for solving large sparse non-Hermitian positive definite system of linear equations. In this paper, we establish a class of multi-step modified Newton-HSS (MMN-HSS) methods for solving large sparse system of nonlinear equations with positive definite Jacobian matrices. The MMN-HSS methods use the multi-step modified Newton methods to solve the nonlinear equations, and the HSS method to approximately solve the Newton equation. Local and semilocal convergence theorems are proved under proper conditions. Also, we present the global multi-step modified Newton-HSS methods with a backtracking strategy and analyse its global convergence. Finally, numerical results are given to confirm the feasibility and effectiveness of our method.