In this article, we introduce a new Steffensen-type method with the advantage that its behavior is very similar to Newton’s method; therefore, it is a very remarkable way of avoiding the drawback that Newton’s method presents for nondifferentiable operators. In our study, we perform an exhaustive comparative study between the semilocal convergence of Newton’s method and the derivative-free point-to-point iterative process considered; in the case of differentiable operators, we use the majoring sequences and the majorant principle. In the nondifferentiable case, we impose conditions on the starting point and on the nonlinear operator to obtain a semilocal convergence result for the iterative process considered. In both cases, we complete the theoretical convergence proofs with a dynamical study and a numerical test. In the case of differentiable operators, this study confirms that the accessibility and numerical behavior of both iterative processes, Newton’s method and the derivative-free point-to-point iterative process considered, are very similar.
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This research was partially supported by Ministerio de Economía y Competitividad under grant PGC2018-095896-B-C21-C22 and by the project EEQ/2018/000720 under Science and Engineering Research Board.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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