Highly efficient family of two-step simultaneous method for all polynomial roots
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 tes...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2024-01-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://aimspress.com/article/doi/10.3934/math.2024085?viewType=HTML |
| _version_ | 1797355401115598848 |
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| author | Mudassir Shams Nasreen Kausar Serkan Araci Liang Kong Bruno Carpentieri |
| author_facet | Mudassir Shams Nasreen Kausar Serkan Araci Liang Kong Bruno Carpentieri |
| author_sort | Mudassir Shams |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T14:10:31Z |
| format | Article |
| id | doaj.art-f47208271dfa4042aec9c785f8dd22d3 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-08T14:10:31Z |
| publishDate | 2024-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-f47208271dfa4042aec9c785f8dd22d32024-01-15T01:17:09ZengAIMS PressAIMS Mathematics2473-69882024-01-01911755177110.3934/math.2024085Highly efficient family of two-step simultaneous method for all polynomial rootsMudassir Shams0Nasreen Kausar1Serkan Araci2Liang Kong3Bruno Carpentieri41. Faculty of Engineering, Free University of Bozen-Bolzano (BZ), 39100, Italy 2. Department of Mathematics and Statistics, Riphah International University Ⅰ-14, Islamabad 44000, Pakistan3. Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Esenler 34220, Istanbul, Türkiye4. Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Türkiye5. Department of Mathematical Sciences and Philosophy, University of Illinois Springfield, USA1. Faculty of Engineering, Free University of Bozen-Bolzano (BZ), 39100, ItalyIn 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.https://aimspress.com/article/doi/10.3934/math.2024085?viewType=HTMLpolynomial equationsnumerical algorithmiterative methodsfractalscpu-time |
| spellingShingle | Mudassir Shams Nasreen Kausar Serkan Araci Liang Kong Bruno Carpentieri Highly efficient family of two-step simultaneous method for all polynomial roots AIMS Mathematics polynomial equations numerical algorithm iterative methods fractals cpu-time |
| title | Highly efficient family of two-step simultaneous method for all polynomial roots |
| title_full | Highly efficient family of two-step simultaneous method for all polynomial roots |
| title_fullStr | Highly efficient family of two-step simultaneous method for all polynomial roots |
| title_full_unstemmed | Highly efficient family of two-step simultaneous method for all polynomial roots |
| title_short | Highly efficient family of two-step simultaneous method for all polynomial roots |
| title_sort | highly efficient family of two step simultaneous method for all polynomial roots |
| topic | polynomial equations numerical algorithm iterative methods fractals cpu-time |
| url | https://aimspress.com/article/doi/10.3934/math.2024085?viewType=HTML |
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