Expanding the applicability of Newton’s method and of a robust modified Newton’s method

Newton’s method cannot be used to approximate a solution of a nonlinear equation when the derivative of the function is singular or almost singular. To overcome this problem a modified Newton’s method may be used. The Newton–Kantorovich theorem is used to show its convergence. The convergence domain of this method is small in general. In the present study, we show how to expand the convergence domain of Newton’s method and the modified Newton’s method by using the center Lipschitz condition and more precise convergence domains than in earlier studies. Numerical examples are also presented.