On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations
In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step <i>p</i>-order method to orde...
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MDPI AG
2024-09-01
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| author | Indra Bate Muniyasamy Murugan Santhosh George Kedarnath Senapati Ioannis K. Argyros Samundra Regmi |
| author_facet | Indra Bate Muniyasamy Murugan Santhosh George Kedarnath Senapati Ioannis K. Argyros Samundra Regmi |
| author_sort | Indra Bate |
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| description | In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step <i>p</i>-order method to order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula>. We obtained the convergence order of Cordero et al.’s method by assuming only up to the third-order derivative of the operator. Our analysis is in a more general commutative Banach algebra setting and provides a radius of the convergence ball. Finally, we validate our theoretical findings with several numerical examples. Also, the concept of basin of attraction is discussed with examples. |
| format | Article |
| id | doaj-art-c818762ac64d46eb8c7b64e0cd167d79 |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-c818762ac64d46eb8c7b64e0cd167d792025-08-20T01:56:00ZengMDPI AGAxioms2075-16802024-09-0113960110.3390/axioms13090601On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear EquationsIndra Bate0Muniyasamy Murugan1Santhosh George2Kedarnath Senapati3Ioannis K. Argyros4Samundra Regmi5Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore 575025, IndiaDepartment of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore 575025, IndiaDepartment of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore 575025, IndiaDepartment of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangalore 575025, IndiaDepartment of Mathematical Sciences, Cameron University, Lawton, OK 73505, USADepartment of Mathematics, University of Houston, Houston, TX 77004, USAIn this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step <i>p</i>-order method to order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula>. We obtained the convergence order of Cordero et al.’s method by assuming only up to the third-order derivative of the operator. Our analysis is in a more general commutative Banach algebra setting and provides a radius of the convergence ball. Finally, we validate our theoretical findings with several numerical examples. Also, the concept of basin of attraction is discussed with examples.https://www.mdpi.com/2075-1680/13/9/601Fréchet derivativeorder of convergenceNewton methodTaylor series expansionnonlinear equationsiterative method |
| spellingShingle | Indra Bate Muniyasamy Murugan Santhosh George Kedarnath Senapati Ioannis K. Argyros Samundra Regmi On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations Axioms Fréchet derivative order of convergence Newton method Taylor series expansion nonlinear equations iterative method |
| title | On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations |
| title_full | On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations |
| title_fullStr | On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations |
| title_full_unstemmed | On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations |
| title_short | On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations |
| title_sort | on extending the applicability of iterative methods for solving systems of nonlinear equations |
| topic | Fréchet derivative order of convergence Newton method Taylor series expansion nonlinear equations iterative method |
| url | https://www.mdpi.com/2075-1680/13/9/601 |
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