Convergence analysis of a class of iterative methods: a unified approach
DOI: https://doi.org/10.3846/mma.2025.21979Abstract
In this paper, we study the convergence of a class of iterative methods for solving the system of nonlinear Banach space valued equations. We provide a unified local and semi-local convergence analysis for these methods. The convergence order of these methods are obtained using the conditions on the derivatives of the involved operator up to order 2 only. Further, we provide the number of iterations required to obtain the given accuracy of the solution. Various numerical examples including integral equations and Caputo fractional differential equations are considered to show the performance of our methods.
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iterative methods, nonlinear equations, Caputo fractional operator, basin of attractionHow to Cite
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Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University.
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References
I.K. Argyros. Convergence and applications of Newton-type iterations. Springer Science & Business Media, 2008.
I.K. Argyros. The theory and applications of iteration methods. CRC Press, 2022. https://doi.org/10.1201/9781003128915
M.K. Argyros, I.K. Argyros, S. Regmi and S. George. Generalized three-step numerical methods for solving equations in banach spaces. Mathematics, 10(15), 2022. https://doi.org/10.3390/math10152621
M.S. Bazaraa, H.D. Sherali and C.M. Shetty. Nonlinear programming: theory and algorithms. John Wiley & Sons, 2013.
B. Campos, J. Canela and P. Vindel. Dynamics of Newton-like root finding methods. Numer. Algorithms, 93(4):1453–1480, 2023. https://doi.org/10.1007/s11075-022-01474-w
J.-L. Chabert, É. Barbin, J. Borowczyk, M. Guillemot and A. Michel-Pajus. A history of algorithms: from the pebble to the microchip. Springer, 1999.
K. Diethelm, N.J. Ford and A.D. Freed. Detailed error analysis for a fractional Adams method. Numer. Algorithms, 36(1):31–52, 2004. https://doi.org/10.1023/B:NUMA.0000027736.85078.be
M. Frontini and E. Sormani. Some variant of Newton’s method with third-order convergence. Appl. Math. Comput., 140(2-3):419–426, 2003. https://doi.org/10.1016/S0096-3003(02)00238-2.
M. Frontini and E. Sormani. Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput., 149(3):771–782, 2004. https://doi.org/10.1016/S0096-3003(03)00178-4
S. George, I. Bate, M. Muniyasamy, G. Chandhini and K. Senapati. Enhancing the applicability of Chebyshev-like method. J. Complexity, 83:101854, 2024. https://doi.org/10.1016/j.jco.2024.101854
M.A. Hernández. Chebyshev’s approximation algorithms and applications. Comput. Math. Appl., 41(3-4):433–445, 2001. https://doi.org/10.1016/S08981221(00)00286-8
H.H.H. Homeier. A modified Newton method with cubic convergence: the multivariate case. J. Comput. Appl. Math., 169(1):161–169, 2004. https://doi.org/10.1016/j.cam.2003.12.041
R. Krishnendu, M. Saeed, S. George and P. Jidesh. On Newton’s midpoint-type iterative scheme’s convergence. Int. J. Appl. Comput. Math., 8(5):266, 2022. https://doi.org/10.1007/s40819-022-01468-1
M. Muniyasamy, G. Chandhini, S. George, I. Bate and K. Senapati. On obtaining convergence order of a fourth and sixth order method of Hueso et al. without using Taylor series expansion. J. Comput. Appl. Math., 452:116136, 2024. https://doi.org/10.1016/j.cam.2024.116136
M. Muniyasamy, G. Santhosh and G. Chandhini. Unified convergence analysis of a class of iterative methods. Numer. Algorithms, 99:683–715, 2025. https://doi.org/10.1007/s11075-024-01893-x
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Copyright (c) 2025 The Author(s). Published by Vilnius Gediminas Technical University.
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This work is licensed under a Creative Commons Attribution 4.0 International License.