In this paper, we present a local convergence analysis of multipoint iterative methods with memory for approximating a locally unique solution of a nonlinear equation. The convergence analysis of these type of methods was shown under hypotheses reaching up to the eighth (or even higher) derivative of the function although only the first derivative appears in the method. The main objective of this study is to expand the applicability of these methods using only hypotheses up to the first derivative of the function. In this way, we extend the applicability of these methods under weaker conditions. Furthermore, the radius of convergence and computable error bounds on the distances involved are also included in this study.
Finally, numerical examples are presented to show that we can solve equations in cases not possible with earlier approaches.

Key words: Local convergence; With memory methods and Computational efficiency.

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