This manuscript is devoted to the study of numerical methods for a class of nonlinear problems. Instead of the standard Newton method, an efficient nonlinear solver is suggested to be used, and it is referred to as the Newton-EHS method, where "EHS" stands for Euler-extrapolated Hermitian-skew-Hermitian splitting. We construct this modified Newton-EHS method by utilizing a modified Newton method as the outer iteration and the EHS method as the inner iteration. Furthermore, we give the derivations of the local and semilocal convergence properties of the proposed method under the Hölder condition. Finally, in order to show the feasibility and validity of our new method, we compare it with some other iterative methods in two numerical examples.

Modified Newton-type methods are efficient for addressing the generalized absolute value equations. In this paper, to further speed up the modified Newton-type methods, two new inexact modified Newton-type iteration methods are proposed. The sufficient conditions for the convergence of the two proposed inexact iteration methods are given. Moreover, to demonstrate the efficacy of the new method, several numerical examples are provided.