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...

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Autor principal:	Mohamed S. M. Bahgat
Formato:	article
Lenguaje:	EN
Publicado:	SpringerOpen 2021
Materias:	Nonlinear equations Free derivative Order of convergence Fractal Basin of attraction Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/b3b3fb50989542d7b3cec1cb66686715
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spelling	oai:doaj.org-article:b3b3fb50989542d7b3cec1cb666867152021-11-08T10:56:35ZThree-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction10.1186/s42787-021-00132-92090-9128https://doaj.org/article/b3b3fb50989542d7b3cec1cb666867152021-11-01T00:00:00Zhttps://doi.org/10.1186/s42787-021-00132-9https://doaj.org/toc/2090-9128Abstract 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.Mohamed S. M. BahgatSpringerOpenarticleNonlinear equationsFree derivativeOrder of convergenceFractalBasin of attractionMathematicsQA1-939ENJournal of the Egyptian Mathematical Society, Vol 29, Iss 1, Pp 1-17 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Nonlinear equations Free derivative Order of convergence Fractal Basin of attraction Mathematics QA1-939
spellingShingle	Nonlinear equations Free derivative Order of convergence Fractal Basin of attraction Mathematics QA1-939 Mohamed S. M. Bahgat Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction
description	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.
format	article
author	Mohamed S. M. Bahgat
author_facet	Mohamed S. M. Bahgat
author_sort	Mohamed S. M. Bahgat
title	Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction
title_short	Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction
title_full	Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction
title_fullStr	Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction
title_full_unstemmed	Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction
title_sort	three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction
publisher	SpringerOpen
publishDate	2021
url	https://doaj.org/article/b3b3fb50989542d7b3cec1cb66686715
work_keys_str_mv	AT mohamedsmbahgat threepointiterativealgorithmintheabsenceofthederivativeforsolvingnonlinearequationsandtheirbasinsofattraction
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Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction

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