Three‑point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction

Document Type : Original Article

Author

Mathematics Department, Faculty of Science, Minia University, Minia, Egypt.

Abstract

In this paper, we suggested and analyzed a new higher-order iterative algorithm for
solving nonlinear equation g(x) = 0, g : R −→ R, which is free from derivative by
using the approximate version of the frst derivative, and we studied the basins of
attraction for the proposed iterative algorithm to fnd complex roots of complex functions g : C −→ C. To show the efectiveness of the proposed algorithm for the real
and the complex domains, the numerical results for the considered examples are given
and graphically clarifed. The basins of attraction of the existing methods and our algorithm are ofered and compared to clarify their performance. The proposed algorithm
satisfed the condition such that |xm − α| < 1.0 × 10−15, as well as the maximum number of iterations is less than or equal to 3, so the proposed algorithm can be applied to
efciently solve numerous type non-linear equations.

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