On an iterative Moser-Kurchatov method for solving systems of nonlinear equations
Abstract
This paper is devoted to analysis of an iterative method for solving nonlinear equations. The method, inspired by the Kurchatov-type methods, is specifically designed to avoid the need for derivative calculations or inverses of linear operators. By employing a sequence of approximating operators and divided differences, the method achieves semilocal convergence. Numerical experiments demonstrate the method’s efficiency and robustness, highlighting its potential advantages over traditional methods like Newton’s method, especially in scenarios where derivative calculations are impractical and computationally expensive. The results indicate that the method is a viable and efficient alternative for solving nonlinear equations, especially in large-scale problems or scenarios, where derivative information is not readily available. The robustness and efficiency of the method make it a valuable tool in various scientific and engineering applications.
References
S. Amat, J.A. Ezquerro, M.A. Hernandez-Veron, Approximation of inverse operators by a new family of high-order iterative methods, Numerical Linear Algebra with Applications, 21 (2014), 629–644. https://doi.org/10.1002/nla.1917
S. Amat, S. Busquier, M.A. Hernandez-Verson, D. Levin, D.F. Yanez, On an iterative Moser-Secant’s method with applications in optimization problems, Preprint (2024). (Private message)
I.K. Argyros, On the Secant method for solving nonsmooth equations, Journal of Mathematical Analysis and Applications, 322 (2006), 146–157. https://doi.org/10.1016/j.jmaa.2005.09.006
I. Argyros, S. Shakhno, Y. Shunkin, Improved convergence analysis of Gauss-Newton-Secant method for solving nonlinear Least Squares Problems, Mathematics, 7 (2019), 99. https://doi.org/10.3390/math7010099
I.K. Argyros, S. Shakhno, Extended Two-Step-Kurchatov method for solving Banach space valued nondifferentiable equations, International Journal of Applied and Computational Mathematics, 6 (2020), 32. https://doi.org/10.1007/s40819-020-0784-y
I.K. Argyros, The theory and applications of iteration methods, 2nd ed., Engineering Series, CRC Press, Taylor & Francis Group, Boca Raton, Florida, USA, 2022. https://doi.org/10.1201/9780203719169
A.S. Berahas, O. Shoham, L.N. Vicente, Full-low evaluation methods for derivative-free optimization, Optimization Methods and Software, 38 (2023), 386–411. https://doi.org/10.48550/arXiv.2107.11908
A. Diaconu, Sur la maniere d’obtenir et sur la convergence de certaines methodes iteratives, Seminar on Functional Analysis and Numerical Methods, Univ. “Babe?-Bolyai”, Cluj-Napoca, Preprint 87-1 (1987), 25–74.
M.J. Ebadi, A. Fahs, H. Fahs, R. Dehghani, Competitive secant (BFGS) methods based on modified secant relations for unconstrained optimization, Optimization, 72 (2023), 1691–1706. https://doi.org/10.1080/02331934.2022.2048381
M. Grau-Sanchez, M. Noguera, S. Amat, On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, Journal of Computational and Applied Mathematics, 237 (2013), 363–372. https://doi.org/10.1016/j.cam.2012.06.005
O.H. Hald, On a Newton-Moser type method, Numerische Mathematik, 23 (1975), 411–425. https://doi.org/10.1007/BF01437039
N. Hamel, N. Benrabia, M. Ghiat, H. Guebbai, A new hybrid conjugate gradient algorithm based on the Newton direction to solve unconstrained optimization problems, Journal of Applied Mathematics and Computing, 69 (2023), 2531–2548. https://doi.org/10.1007/s12190-022-01821-z
M.A. Hernandez-Veron, A.A. Magzenuin, E. Munez, E.G. Villalba, Solving non-differentiable Hammerstein integral equations via first-order divided differences, Numerical Algorithms, 97 (2024), 567–594. https://doi.org/10.1007/s11075-023-01715-6
I. Moret, A note on Newton type iterative methods, Computing, 33 (1984), 65–73. https://doi.org/10.1007/BF02243076
M.A. Nawkhas, N.A. Sulaiman, Using geometric arithmetic mean to solve non-linear fractional programming problems, International Journal of Applied and Computational Mathematics, 8 (2022), 217. https://doi.org/10.1007/s40819-022-01424-z
Y. Nesterov, Superfast second-order methods for unconstrained convex optimization, Journal of Optimization Theory and Applications, 191 (2021), 1–30. https://doi.org/10.1007/s10957-021-01930-y
H. Petzeltova, Remark on a Newton-Moser type method, Commentationes Mathematicae Universitatis Carolinae, 21 (1980), 719–725.
S.M. Shakhno, Nonlinear majorants for investigation of methods of linear interpolation for the solution of nonlinear equations, ECCOMAS 2004—European Congress on Computational Methods in Applied Sciences and Engineering, 2004.
S.M. Shakhno, Gauss–Newton–Kurchatov Method for the solution of nonlinear Least-Squares Problems, Journal of Mathematical Sciences, 247 (2020), 58–72. https://doi.org/10.1007/s10958-020-04789-y
S. Ulm, Ob obobshchennykh razdelennykh raznostyakh, Eesti NSV Teaduste Akadeemia Toimetised Fuusika-Matemaatika, 16 (1967), 146–156.
T. Wang, Q. Huang, A new Newton method for convex optimization problems with singular Hessian matrices, AIMS Mathematics, 8 (2023), 21161–21175. https://doi.org/10.3934/math.20231078
G.L. Yuan, Y.-J. Zhou, M.-X. Zhang, A hybrid conjugate gradient algorithm for nonconvex functions and its applications in image restoration problems, Journal of the Operations Research Society of China, 11 (2023), 759–781. https://doi.org/10.1007/s40305-022-00424-6
E.J. Zehnder, A remark about Newton’s method, Communications on Pure and Applied Mathematics, 27 (1974), 361–366. https://doi.org/10.1002/cpa.3160270305
Copyright (c) 2025 S. M. Shakhno, I. K. Argyros, Y. V. Shunkin
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.
