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Ball convergence for a derivative free method with memory for solving nonlinear equations

Abstract:

In this paper, we present a local convergence analysis of a derivative free methods of order 1 + √2in order to obtain a locally unique solution in a Banach space setting. Further, we use hypotheses only on the first order derivative of the involved function. On the other hand, in the earlier study Compas et al. [7] proved the order of convergence with help of Taylor series expansions and hypotheses up to the third order derivative or even higher of the function involved which put restriction on the applicability of the proposed scheme. However, the proposed scheme is derivative free. The principle aim of this study is not only to expand the applicability of the proposed scheme on those examples where earlier study fails to work but also expand the space of application and proposed the convergence domain. Moreover, a variety of concrete numerical examples are demonstrated to confirm the proposed theory in this paper.