On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations
Abstract
:1. Introduction
- (i)
- We obtain the convergence order of method (3) without using the Taylor series expansion.
- (ii)
- We use conditions on the operator and its Fréchet derivates up to the third order only.
- (iii)
- We provide the convergence ball which was not given in earlier studies.
- (iv)
- The analysis in [10] was conducted in the Euclidean spaces, whereas our study is in a more general commutative Banach algebra setting.
2. Preliminary Concepts
- (i)
- ,
- (ii)
- There exist , , and such that
- (C1)
- (C2)
- ;
- (C3)
- ;
- (C4)
- ;
- (C5)
- ;
- (C6)
- .
3. Main Results
4. Numerical Examples
5. Basin of Attractions
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Grids | as in (18) | as in (19) | as in (20) | |||
---|---|---|---|---|---|---|
0 | 0 | 0 | ||||
0 | 0 | 0 | ||||
0 | 0 | 0 | ||||
0 | 0 | 0 | ||||
0 | 0 | 0 | ||||
0 | 0 | 0 | ||||
0 | 0 | 0 | ||||
0 | 0 | 0 |
Iteration | as in (18) | as in (19) | as in (20) |
---|---|---|---|
1 | (, 0) | (, 0) | (, 0) |
2 | (, 0) | (0, , 0) | (0, 0, 0) |
3 | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) |
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Bate, I.; Murugan, M.; George, S.; Senapati, K.; Argyros, I.K.; Regmi, S. On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations. Axioms 2024, 13, 601. https://doi.org/10.3390/axioms13090601
Bate I, Murugan M, George S, Senapati K, Argyros IK, Regmi S. On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations. Axioms. 2024; 13(9):601. https://doi.org/10.3390/axioms13090601
Chicago/Turabian StyleBate, Indra, Muniyasamy Murugan, Santhosh George, Kedarnath Senapati, Ioannis K. Argyros, and Samundra Regmi. 2024. "On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations" Axioms 13, no. 9: 601. https://doi.org/10.3390/axioms13090601