Local convergence for a derivative free method of order three under weak conditions

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A local convergence analysis for a family of a third-order method in order to approximate a solution of a nonlinear equation is presented in this paper. We use hypotheses only on the first derivative in contrast to earlier studies such as Parhi and Gupta (2007, 2010) and Zhu and Wu (2003) using hypotheses only on the first derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.