Convergence analysis of modified Newton‐HSS method for solving systems of nonlinear equations

Hermitian and skew‐Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non‐Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton‐HSS method for solving the systems of nonlinear equations with non‐Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton‐HSS method outperforms the Newton‐USOR and the Newton‐GMRES iteration methods. In this paper, a class of modified Newton‐HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R‐order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton‐HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.