Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms
In the study of systems’ dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local sy...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/6/1249 |
| _version_ | 1797592521415589888 |
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| author | Sunil Kumar Janak Raj Sharma Jai Bhagwan Lorentz Jäntschi |
| author_facet | Sunil Kumar Janak Raj Sharma Jai Bhagwan Lorentz Jäntschi |
| author_sort | Sunil Kumar |
| collection | DOAJ |
| description | In the study of systems’ dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local symmetry, dictated the dynamics of gravity, of space-time itself. In certain instances, especially in the presence of symmetry, we end up having to deal with an equation with multiple roots. A variety of optimal methods have been proposed in the literature for multiple roots with known multiplicity, all of which need derivative evaluations in the formulations. However, in the literature, optimal methods without derivatives are few. Motivated by this feature, here we present a novel optimal family of fourth-order methods for multiple roots with known multiplicity, which do not use any derivative. The scheme of the new iterative family consists of two steps, namely Traub-Steffensen and Traub-Steffensen-like iterations with weight factor. According to the Kung-Traub hypothesis, the new algorithms satisfy the optimality criterion. Taylor’s series expansion is used to examine order of convergence. We also demonstrate the application of new algorithms to real-life problems, i.e., Van der Waals problem, Manning problem, Planck law radiation problem, and Kepler’s problem. Furthermore, the performance comparisons have shown that the given derivative-free algorithms are competitive with existing optimal fourth-order algorithms that require derivative information. |
| first_indexed | 2024-03-11T01:53:16Z |
| format | Article |
| id | doaj.art-9a7ff19a41a64e6e815bf9423a62dc1c |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T01:53:16Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-9a7ff19a41a64e6e815bf9423a62dc1c2023-11-18T12:51:33ZengMDPI AGSymmetry2073-89942023-06-01156124910.3390/sym15061249Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free AlgorithmsSunil Kumar0Janak Raj Sharma1Jai Bhagwan2Lorentz Jäntschi3Department of Mathematics, University Centre for Research and Development, Chandigarh University, Mohali 140413, Punjab, IndiaDepartment of Mathematics, Sant Longowal Institute of Engineering Technology, Longowal 148106, Punjab, IndiaDepartment of Mathematics, Pt. NRS Government College, Rohtak 124001, Haryana, IndiaDepartment of Physics and Chemistry, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, RomaniaIn the study of systems’ dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local symmetry, dictated the dynamics of gravity, of space-time itself. In certain instances, especially in the presence of symmetry, we end up having to deal with an equation with multiple roots. A variety of optimal methods have been proposed in the literature for multiple roots with known multiplicity, all of which need derivative evaluations in the formulations. However, in the literature, optimal methods without derivatives are few. Motivated by this feature, here we present a novel optimal family of fourth-order methods for multiple roots with known multiplicity, which do not use any derivative. The scheme of the new iterative family consists of two steps, namely Traub-Steffensen and Traub-Steffensen-like iterations with weight factor. According to the Kung-Traub hypothesis, the new algorithms satisfy the optimality criterion. Taylor’s series expansion is used to examine order of convergence. We also demonstrate the application of new algorithms to real-life problems, i.e., Van der Waals problem, Manning problem, Planck law radiation problem, and Kepler’s problem. Furthermore, the performance comparisons have shown that the given derivative-free algorithms are competitive with existing optimal fourth-order algorithms that require derivative information.https://www.mdpi.com/2073-8994/15/6/1249multiple rootsconvergencenonlinear equationsderivative-free method |
| spellingShingle | Sunil Kumar Janak Raj Sharma Jai Bhagwan Lorentz Jäntschi Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms Symmetry multiple roots convergence nonlinear equations derivative-free method |
| title | Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms |
| title_full | Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms |
| title_fullStr | Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms |
| title_full_unstemmed | Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms |
| title_short | Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms |
| title_sort | numerical solution of nonlinear problems with multiple roots using derivative free algorithms |
| topic | multiple roots convergence nonlinear equations derivative-free method |
| url | https://www.mdpi.com/2073-8994/15/6/1249 |
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