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

Abstract

In this paper, we suggested and analyzed a new higher-order iterative algorithm for solving nonlinear equation $$g(x)=0$$ g ( x ) = 0 , $$g:{\mathbb {R}}\longrightarrow {\mathbb {R}}$$ g : R ⟶ R , which is free from derivative by using the approximate version of the first derivative, and we studied the basins of attraction for the proposed iterative algorithm to find complex roots of complex functions $$g:{\mathbb {C}}\longrightarrow {\mathbb {C}}$$ g : C ⟶ C . To show the effectiveness of the proposed algorithm for the real and the complex domains, the numerical results for the considered examples are given and graphically clarified. The basins of attraction of the existing methods and our algorithm are offered and compared to clarify their performance. The proposed algorithm satisfied the condition such that $$|x_{m}-\alpha |<1.0 \times 10^{-15}$$ | x m - 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 efficiently solve numerous type non-linear equations.

An Efficient Derivative Free Iterative Method and Condition Number of Nonlinear Equations Arising from Real World Phenomena

“…The convergence analysis of Algorithm 1 closely aligns with the findings of Bahgat (2021). Numerical experiments conducted by Bahgat (2021) demonstrate that Algorithm 1 yields commendable performance, exhibiting minimal error when 𝛽 = 1 .…”

“…The convergence analysis of Algorithm 1 closely aligns with the findings of Bahgat (2021). Numerical experiments conducted by Bahgat (2021) demonstrate that Algorithm 1 yields commendable performance, exhibiting minimal error when 𝛽 = 1 . This level of accuracy is well within our prescribed tolerance and comparable to the results obtained with 𝛽 = −1 or 𝛽 = −0.5 .…”

“…This research work, therefore, aimed to make the following contributions to the field. Firstly, it proposes Algorithm 2, which is a non-derivative ITM, as one of the most efficient method for solving NLE when compared to its variant method presented by Bahgat (2021). Secondly, the notion of CN of NLE arising from real world phenomena is introduced.…”