Convergence analysis of the modified Newton–HSS method under the Hölder continuous condition

Abstract

The modified Newton–HSS method, which is constructed by employing the Hermitian and skew-Hermitian splitting methods as the inner iteration process at each step of the outer modified Newton’s iteration, has been proved to be a competitive method for solving large sparse systems of nonlinear equations with non-Hermitian positive-definite Jacobian matrices. In this paper, under the hypotheses that the derivative is continuous and the derivative satisfies the Hölder continuous condition, two local convergence theorems are established for the modified Newton–HSS method. Furthermore, the rate of convergence of the modified Newton–HSS method is also characterized in terms of the rate of convergence of the matrix ‖T(α;x)‖. The numerical example is given to confirm the concrete applications of the results of our paper.