Highly efficient family of two-step simultaneous method for all polynomial roots

Mudassir Shams; Nasreen Kausar; Serkan Araci; Liang Kong; Bruno Carpentieri; Mudassir Shams; Nasreen Kausar; Serkan Araci; Liang Kong; Bruno Carpentieri

doi:10.3934/math.2024085

AIMS Mathematics

2024, Volume 9, Issue 1: 1755-1771. doi: 10.3934/math.2024085

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Highly efficient family of two-step simultaneous method for all polynomial roots

1.
Faculty of Engineering, Free University of Bozen-Bolzano (BZ), 39100, Italy
2.
Department of Mathematics and Statistics, Riphah International University Ⅰ-14, Islamabad 44000, Pakistan
3.
Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Esenler 34220, Istanbul, Türkiye
4.
Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Türkiye
5.
Department of Mathematical Sciences and Philosophy, University of Illinois Springfield, USA

Received: 29 September 2023 Revised: 07 November 2023 Accepted: 07 November 2023 Published: 14 December 2023
MSC : 65H04, 65H05, 65Y05, 65M12

In this article, we constructed a derivative-free family of iterative techniques for extracting simultaneously all the distinct roots of a non-linear polynomial equation. Convergence analysis is discussed to show that the proposed family of iterative method has fifth order convergence. Nonlinear test models including fractional conversion, predator-prey, chemical reactor and beam designing models are included. Also many other interesting results concerning symmetric problems with application of group symmetry are also described. The simultaneous iterative scheme is applied starting with the initial estimates to get the exact roots within the given tolerance. The proposed iterative scheme requires less function evaluations and computation time as compared to existing classical methods. Dynamical planes are exhibited in CAS-MATLAB (R2011B) to show how the simultaneous iterative approach outperforms single roots finding methods that might confine the divergence zone in terms of global convergence. Furthermore, convergence domains, namely basins of attraction that are symmetrical through fractal-like edges, are analyzed using the graphical tool. Numerical results and residual graphs are presented in detail for the simultaneous iterative method. An extensive study has been made for the newly developed simultaneous iterative scheme, which is found to be efficient, robust and authentic in its domain.
- polynomial equations,
- numerical algorithm,
- iterative methods,
- fractals,
- CPU-time
Citation: Mudassir Shams, Nasreen Kausar, Serkan Araci, Liang Kong, Bruno Carpentieri. Highly efficient family of two-step simultaneous method for all polynomial roots[J]. AIMS Mathematics, 2024, 9(1): 1755-1771. doi: 10.3934/math.2024085

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Abstract

In this article, we constructed a derivative-free family of iterative techniques for extracting simultaneously all the distinct roots of a non-linear polynomial equation. Convergence analysis is discussed to show that the proposed family of iterative method has fifth order convergence. Nonlinear test models including fractional conversion, predator-prey, chemical reactor and beam designing models are included. Also many other interesting results concerning symmetric problems with application of group symmetry are also described. The simultaneous iterative scheme is applied starting with the initial estimates to get the exact roots within the given tolerance. The proposed iterative scheme requires less function evaluations and computation time as compared to existing classical methods. Dynamical planes are exhibited in CAS-MATLAB (R2011B) to show how the simultaneous iterative approach outperforms single roots finding methods that might confine the divergence zone in terms of global convergence. Furthermore, convergence domains, namely basins of attraction that are symmetrical through fractal-like edges, are analyzed using the graphical tool. Numerical results and residual graphs are presented in detail for the simultaneous iterative method. An extensive study has been made for the newly developed simultaneous iterative scheme, which is found to be efficient, robust and authentic in its domain.

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