Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations
Chebyshev-type methods have replaced the Chebyshev method in practice for solving nonlinear equations in abstract spaces. These methods are of the same R-order of three. However, they are easier to deal with, since the computationally expensive second derivative of the operator involved does not app...
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MDPI AG
2024-12-01
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| author | Ioannis K. Argyros Santhosh George |
| author_facet | Ioannis K. Argyros Santhosh George |
| author_sort | Ioannis K. Argyros |
| collection | DOAJ |
| description | Chebyshev-type methods have replaced the Chebyshev method in practice for solving nonlinear equations in abstract spaces. These methods are of the same R-order of three. However, they are easier to deal with, since the computationally expensive second derivative of the operator involved does not appear on these methods. However, the invertibility of the first derivative is still required at each step of the iteration. In this article, the inverse is replaced by a finite sum of linear operators. The convergence of the new Hybrid Chebyshev-Type Method (HCTM) is established under relaxed generalized continuity assumptions on the derivative and majorizing sequences. The iterates of the new methods converge to the original ones, but they are easier to find. Moreover, the numerical examples demonstrate that the new iterates converge essentially as fast to the solution. The methodology of this article can be used on other methods with inverses along the same lines due to its generality. |
| format | Article |
| id | doaj-art-7d1f07eea78a4f28bca6124a1231222e |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-7d1f07eea78a4f28bca6124a1231222e2025-01-10T13:18:10ZengMDPI AGMathematics2227-73902024-12-011317410.3390/math13010074Hybrid Chebyshev-Type Methods for Solving Nonlinear EquationsIoannis K. Argyros0Santhosh George1Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USADepartment of Mathematical & Computational Science, National Institute of Technology Karnataka, Surathkal, Mangaluru 575 025, IndiaChebyshev-type methods have replaced the Chebyshev method in practice for solving nonlinear equations in abstract spaces. These methods are of the same R-order of three. However, they are easier to deal with, since the computationally expensive second derivative of the operator involved does not appear on these methods. However, the invertibility of the first derivative is still required at each step of the iteration. In this article, the inverse is replaced by a finite sum of linear operators. The convergence of the new Hybrid Chebyshev-Type Method (HCTM) is established under relaxed generalized continuity assumptions on the derivative and majorizing sequences. The iterates of the new methods converge to the original ones, but they are easier to find. Moreover, the numerical examples demonstrate that the new iterates converge essentially as fast to the solution. The methodology of this article can be used on other methods with inverses along the same lines due to its generality.https://www.mdpi.com/2227-7390/13/1/74Chebyshev methodoptimized and hybrid Chebyshev-type methodsBanach spaceconvergenceinverse of an operator |
| spellingShingle | Ioannis K. Argyros Santhosh George Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations Mathematics Chebyshev method optimized and hybrid Chebyshev-type methods Banach space convergence inverse of an operator |
| title | Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations |
| title_full | Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations |
| title_fullStr | Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations |
| title_full_unstemmed | Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations |
| title_short | Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations |
| title_sort | hybrid chebyshev type methods for solving nonlinear equations |
| topic | Chebyshev method optimized and hybrid Chebyshev-type methods Banach space convergence inverse of an operator |
| url | https://www.mdpi.com/2227-7390/13/1/74 |
| work_keys_str_mv | AT ioanniskargyros hybridchebyshevtypemethodsforsolvingnonlinearequations AT santhoshgeorge hybridchebyshevtypemethodsforsolvingnonlinearequations |