Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction
Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore,...
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MDPI AG
2019-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/11/1052 |
| _version_ | 1818300105620455424 |
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| author | Jian Li Xiaomeng Wang Kalyanasundaram Madhu |
| author_facet | Jian Li Xiaomeng Wang Kalyanasundaram Madhu |
| author_sort | Jian Li |
| collection | DOAJ |
| description | Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore, all these algorithms are optimal in the sense of Kung−Traub conjecture; the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have given convergence analyses of the proposed methods and also given comparisons with already established known schemes having the same convergence order, demonstrating the efficiency of the present techniques numerically. We also studied basins of attraction to demonstrate their dynamical behavior in the complex plane. |
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| format | Article |
| id | doaj.art-9eeafdd855e34bc5991c4cbe4722de12 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-13T05:01:50Z |
| publishDate | 2019-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9eeafdd855e34bc5991c4cbe4722de122022-12-21T23:58:46ZengMDPI AGMathematics2227-73902019-11-01711105210.3390/math7111052math7111052Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of AttractionJian Li0Xiaomeng Wang1Kalyanasundaram Madhu2Inner Mongolia Vocational College of Chemical Engineering, Hohhot 010070, ChinaInner Mongolia Vocational College of Chemical Engineering, Hohhot 010070, ChinaDepartment of Mathematics, Saveetha Engineering College, Chennai 602105, IndiaBased on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore, all these algorithms are optimal in the sense of Kung−Traub conjecture; the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have given convergence analyses of the proposed methods and also given comparisons with already established known schemes having the same convergence order, demonstrating the efficiency of the present techniques numerically. We also studied basins of attraction to demonstrate their dynamical behavior in the complex plane.https://www.mdpi.com/2227-7390/7/11/1052kung–traub conjecturemultipoint iterationsnonlinear equationoptimal orderbasins of attraction |
| spellingShingle | Jian Li Xiaomeng Wang Kalyanasundaram Madhu Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction Mathematics kung–traub conjecture multipoint iterations nonlinear equation optimal order basins of attraction |
| title | Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction |
| title_full | Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction |
| title_fullStr | Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction |
| title_full_unstemmed | Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction |
| title_short | Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction |
| title_sort | higher order derivative free iterative methods for solving nonlinear equations and their basins of attraction |
| topic | kung–traub conjecture multipoint iterations nonlinear equation optimal order basins of attraction |
| url | https://www.mdpi.com/2227-7390/7/11/1052 |
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