The cubic semilocal convergence on two variants of Newtonʹs method

In this paper, we discuss two variants of Newtonʹs method without using any second derivative for solving nonlinear equations. By using the majorant function and confirming the majorant sequences, we obtain the cubic semilocal convergence and the error estimation in the Kantorovich-type theorems. The numerical examples are presented to support the usefulness and significance.