On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space
A plethora of methods are used for solving equations in the finite-dimensional Euclidean space. Higher-order derivatives, on the other hand, are utilized in the calculation of the local convergence order. However, these derivatives are not on the methods. Moreover, no bounds on the error and uniquen...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/16/1/2 |
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| author | Ioannis K. Argyros Stepan Shakhno Samundra Regmi Halyna Yarmola |
| author_facet | Ioannis K. Argyros Stepan Shakhno Samundra Regmi Halyna Yarmola |
| author_sort | Ioannis K. Argyros |
| collection | DOAJ |
| description | A plethora of methods are used for solving equations in the finite-dimensional Euclidean space. Higher-order derivatives, on the other hand, are utilized in the calculation of the local convergence order. However, these derivatives are not on the methods. Moreover, no bounds on the error and uniqueness information for the solution are given either. Thus, the advantages of these methods are restricted in their application to equations with operators that are sufficiently many times differentiable. These limitations motivate us to write this paper. In particular, we present the more interesting semi-local convergence analysis not given previously for two sixth-order methods that are run under the same set of conditions. The technique is based on the first derivative that only appears in the methods. This way, these methods are more applicable for addressing equations and in the more general setting of Banach space-valued operators. Hence, the applicability is extended for these methods. This is the novelty of the paper. The same technique can be used in other methods. Finally, examples are used to test the convergence of the methods. |
| first_indexed | 2024-03-09T13:49:53Z |
| format | Article |
| id | doaj.art-2fd3baf3a654464a839a0b81fe59c59c |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-03-09T13:49:53Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-2fd3baf3a654464a839a0b81fe59c59c2023-11-30T20:51:02ZengMDPI AGAlgorithms1999-48932022-12-01161210.3390/a16010002On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach SpaceIoannis K. Argyros0Stepan Shakhno1Samundra Regmi2Halyna Yarmola3Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USADepartment of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, UkraineDepartment of Mathematics, University of Houston, Houston, TX 77204, USADepartment of Computational Mathematics, Ivan Franko National University of Lviv, Universytetska Str. 1, 79000 Lviv, UkraineA plethora of methods are used for solving equations in the finite-dimensional Euclidean space. Higher-order derivatives, on the other hand, are utilized in the calculation of the local convergence order. However, these derivatives are not on the methods. Moreover, no bounds on the error and uniqueness information for the solution are given either. Thus, the advantages of these methods are restricted in their application to equations with operators that are sufficiently many times differentiable. These limitations motivate us to write this paper. In particular, we present the more interesting semi-local convergence analysis not given previously for two sixth-order methods that are run under the same set of conditions. The technique is based on the first derivative that only appears in the methods. This way, these methods are more applicable for addressing equations and in the more general setting of Banach space-valued operators. Hence, the applicability is extended for these methods. This is the novelty of the paper. The same technique can be used in other methods. Finally, examples are used to test the convergence of the methods.https://www.mdpi.com/1999-4893/16/1/2Banach spacesFréchet derivativeconvergence ordersemi-local convergenceconvergence ball |
| spellingShingle | Ioannis K. Argyros Stepan Shakhno Samundra Regmi Halyna Yarmola On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space Algorithms Banach spaces Fréchet derivative convergence order semi-local convergence convergence ball |
| title | On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space |
| title_full | On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space |
| title_fullStr | On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space |
| title_full_unstemmed | On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space |
| title_short | On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space |
| title_sort | on the semi local convergence of two competing sixth order methods for equations in banach space |
| topic | Banach spaces Fréchet derivative convergence order semi-local convergence convergence ball |
| url | https://www.mdpi.com/1999-4893/16/1/2 |
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