A semi-local convergence theorem for a robust revised Newtonʹs method

It is well known that Newtonʹs iteration will abort due to the overflow if the derivative of the function at an iterate is singular or almost singular. In this paper, we study a robust revised Newtonʹs method for solving nonlinear equations, which can be carried out with a starting point with a degenerate derivative at an iterative step. It is proved that the method is convergent under the conditions of the Newton Kantorovich theorem, which implies a larger convergence domain of the method. We also show that our method inherits the fast convergence of Newtonʹs method. Numerical experiments are performed to show the robustness of the proposed method in comparison with the standard Newtonʹs method.