On generalized one-step derivative-free iterative family for evaluating multiple roots
In this study, we propose a family of iterative procedures with no deriva-tives for calculating multiple roots of one-variable nonlinear equations. We also present an iterative technique to approximate the multiplicity of the roots. The new class is optimal since it fits the Kung–Traub hypothesis an...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Ferdowsi University of Mashhad
2024-03-01
|
| Series: | Iranian Journal of Numerical Analysis and Optimization |
| Subjects: | |
| Online Access: | https://ijnao.um.ac.ir/article_44485_b3cf3bd3a48aa8ab717883ce3d827566.pdf |
| _version_ | 1797296206519468032 |
|---|---|
| author | H. Arora A. Cordero J. R. Torregrosa |
| author_facet | H. Arora A. Cordero J. R. Torregrosa |
| author_sort | H. Arora |
| collection | DOAJ |
| description | In this study, we propose a family of iterative procedures with no deriva-tives for calculating multiple roots of one-variable nonlinear equations. We also present an iterative technique to approximate the multiplicity of the roots. The new class is optimal since it fits the Kung–Traub hypothesis and has second-order convergence. Derivative-free methods for calculating mul-tiple roots are rarely found in literature, especially in the case of one-step methods, which are the simplest ones in terms of their structure. Moreover, this new family contains almost all the existing single-step derivative-free iterative schemes as its special cases, with an additional degree of freedom. Several results are used to confirm its theoretical order of convergence. Through the complex discrete dynamics analysis, the stability of the sug-gested class is illustrated, and the most stable methods are found. Several test problems are included to check the performance of the proposed meth-ods, whether the multiplicity of the roots is estimated or known, comparing the numerical results with those obtained by other methods. |
| first_indexed | 2024-03-07T22:00:01Z |
| format | Article |
| id | doaj.art-5dbb77b9a95e4cb9b19411f178164bc7 |
| institution | Directory Open Access Journal |
| issn | 2423-6977 2423-6969 |
| language | English |
| last_indexed | 2024-03-07T22:00:01Z |
| publishDate | 2024-03-01 |
| publisher | Ferdowsi University of Mashhad |
| record_format | Article |
| series | Iranian Journal of Numerical Analysis and Optimization |
| spelling | doaj.art-5dbb77b9a95e4cb9b19411f178164bc72024-02-24T05:13:09ZengFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-69772423-69692024-03-0114Issue 129131410.22067/ijnao.2023.82958.128244485On generalized one-step derivative-free iterative family for evaluating multiple rootsH. Arora0A. Cordero1J. R. Torregrosa2Department of Mathematics, Guru Nanak Dev University, 143005, Amritsar, India.Instituto de Matemàtica Multidisciplinar, Universitat Politècnica de València, 46022, Valencia, Spain.Instituto de Matemàtica Multidisciplinar, Universitat Politècnica de València, 46022, Valencia, Spain.In this study, we propose a family of iterative procedures with no deriva-tives for calculating multiple roots of one-variable nonlinear equations. We also present an iterative technique to approximate the multiplicity of the roots. The new class is optimal since it fits the Kung–Traub hypothesis and has second-order convergence. Derivative-free methods for calculating mul-tiple roots are rarely found in literature, especially in the case of one-step methods, which are the simplest ones in terms of their structure. Moreover, this new family contains almost all the existing single-step derivative-free iterative schemes as its special cases, with an additional degree of freedom. Several results are used to confirm its theoretical order of convergence. Through the complex discrete dynamics analysis, the stability of the sug-gested class is illustrated, and the most stable methods are found. Several test problems are included to check the performance of the proposed meth-ods, whether the multiplicity of the roots is estimated or known, comparing the numerical results with those obtained by other methods.https://ijnao.um.ac.ir/article_44485_b3cf3bd3a48aa8ab717883ce3d827566.pdfnonlinear equationsderivative–free iterative methodmultiple rootsorder of convergencestability |
| spellingShingle | H. Arora A. Cordero J. R. Torregrosa On generalized one-step derivative-free iterative family for evaluating multiple roots Iranian Journal of Numerical Analysis and Optimization nonlinear equations derivative–free iterative method multiple roots order of convergence stability |
| title | On generalized one-step derivative-free iterative family for evaluating multiple roots |
| title_full | On generalized one-step derivative-free iterative family for evaluating multiple roots |
| title_fullStr | On generalized one-step derivative-free iterative family for evaluating multiple roots |
| title_full_unstemmed | On generalized one-step derivative-free iterative family for evaluating multiple roots |
| title_short | On generalized one-step derivative-free iterative family for evaluating multiple roots |
| title_sort | on generalized one step derivative free iterative family for evaluating multiple roots |
| topic | nonlinear equations derivative–free iterative method multiple roots order of convergence stability |
| url | https://ijnao.um.ac.ir/article_44485_b3cf3bd3a48aa8ab717883ce3d827566.pdf |
| work_keys_str_mv | AT harora ongeneralizedonestepderivativefreeiterativefamilyforevaluatingmultipleroots AT acordero ongeneralizedonestepderivativefreeiterativefamilyforevaluatingmultipleroots AT jrtorregrosa ongeneralizedonestepderivativefreeiterativefamilyforevaluatingmultipleroots |