An improved bisection Newton-like method for enclosing simple zeros of nonlinear equations

Abstract

This paper is devoted to propose a new bisection Newton-like method of order four for enclosing simple zero \(\alpha \) of nonlinear equations. The method is a combination of a derivative free Newton-like method of order four and bisection method. Starting with a suitably chosen initial approximation \(x_{0}\in [a_{0},b_{0}]\), the method generates sequence of successive iterates {\(x_{n}\)} and sequence of intervals {\([a_{n},b_{n}]\)} containing zeros. Convergence analysis for the method has also been carried out and showed that sequence of diameters {\((b_{n}-a_{n})\)} and sequence of errors {\((x_{n}-\alpha )\)} converges to 0 simultaneously with fourth order of convergence. A number of numerical examples are also included and compared our results with other similar results.