An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics
In this paper, we study the convergence and complex dynamics of a novel higher-order multipoint iteration scheme to solve nonlinear equations. The approach is based upon utilizing cubic interpolation in the second step of the King–Werner method to improve its convergence order from <inline-formul...
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2025-05-01
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| author | Moin-ud-Din Junjua Ibraheem M. Alsulami Amer Alsulami Sangeeta Kumari |
| author_facet | Moin-ud-Din Junjua Ibraheem M. Alsulami Amer Alsulami Sangeeta Kumari |
| author_sort | Moin-ud-Din Junjua |
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| description | In this paper, we study the convergence and complex dynamics of a novel higher-order multipoint iteration scheme to solve nonlinear equations. The approach is based upon utilizing cubic interpolation in the second step of the King–Werner method to improve its convergence order from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2.414</mn></mrow></semantics></math></inline-formula> to 3 and the efficiency index from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.554</mn></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.732</mn></mrow></semantics></math></inline-formula>, which is higher than the efficiency of optimal fourth- and eighth-order iterative schemes. The proposed method is validated through numerical and dynamic experiments concerning the absolute error, approximated computational order, regions of convergence, and CPU time (sec) on the real-world problems, including Kepler’s equation, isentropic supersonic flow, and law of population growth, demonstrating superior performance compared to some existing well-known methods. Commonly, regions of convergence of iterative methods are investigated and compared by plotting attractor basins of iteration schemes in the complex plane on polynomial functions of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>z</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. However, in this paper, the attractor basins of the proposed method are investigated on diverse nonlinear functions. The proposed scheme creates portraits of basins of attraction faster with wider convergence areas outperforming existing well-known iteration schemes. |
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| language | English |
| publishDate | 2025-05-01 |
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| spelling | doaj-art-37a62b2f84d645508f475cbbed6893bd2025-08-20T03:14:32ZengMDPI AGAxioms2075-16802025-05-0114536010.3390/axioms14050360An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex DynamicsMoin-ud-Din Junjua0Ibraheem M. Alsulami1Amer Alsulami2Sangeeta Kumari3School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, Zhejiang, ChinaMathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi ArabiaDepartment of Mathematics, Turabah University College, Taif University, Taif 21944, Saudi ArabiaDepartment of Mathematics, Chandigarh University, Gharuan, Mohali 140301, IndiaIn this paper, we study the convergence and complex dynamics of a novel higher-order multipoint iteration scheme to solve nonlinear equations. The approach is based upon utilizing cubic interpolation in the second step of the King–Werner method to improve its convergence order from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2.414</mn></mrow></semantics></math></inline-formula> to 3 and the efficiency index from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.554</mn></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1.732</mn></mrow></semantics></math></inline-formula>, which is higher than the efficiency of optimal fourth- and eighth-order iterative schemes. The proposed method is validated through numerical and dynamic experiments concerning the absolute error, approximated computational order, regions of convergence, and CPU time (sec) on the real-world problems, including Kepler’s equation, isentropic supersonic flow, and law of population growth, demonstrating superior performance compared to some existing well-known methods. Commonly, regions of convergence of iterative methods are investigated and compared by plotting attractor basins of iteration schemes in the complex plane on polynomial functions of the type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>z</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>. However, in this paper, the attractor basins of the proposed method are investigated on diverse nonlinear functions. The proposed scheme creates portraits of basins of attraction faster with wider convergence areas outperforming existing well-known iteration schemes.https://www.mdpi.com/2075-1680/14/5/360nonlinear equationsKing–Werner type methodR-order of convergenceefficiency indexcomplex dynamics |
| spellingShingle | Moin-ud-Din Junjua Ibraheem M. Alsulami Amer Alsulami Sangeeta Kumari An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics Axioms nonlinear equations King–Werner type method R-order of convergence efficiency index complex dynamics |
| title | An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics |
| title_full | An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics |
| title_fullStr | An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics |
| title_full_unstemmed | An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics |
| title_short | An Improved King–Werner-Type Method Based on Cubic Interpolation: Convergence Analysis and Complex Dynamics |
| title_sort | improved king werner type method based on cubic interpolation convergence analysis and complex dynamics |
| topic | nonlinear equations King–Werner type method R-order of convergence efficiency index complex dynamics |
| url | https://www.mdpi.com/2075-1680/14/5/360 |
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