Extended local convergence for Newton-type solver under weak conditions
DOI:
https://doi.org/10.24193/subbmath.2021.4.12Keywords:
Banach space, Newton-type, local convergence, Fr´echet derivative.Abstract
We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k−dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the first derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented.
Mathematics Subject Classification (2010): 65F08, 37F50, 65N12.
References
Argyros, I.K., Ezquerro, J.A., Gutierrez, J.M., Hernandez, M.A., Hilout, S., On the semilocal convergence of efficient Chebyshev-Secant-type methods, J. Comput. Appl. Math., 235(2011), 3195-3206.
Argyros, I.K., George, S., Thapa, N., Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications, Vol. I, Nova Publishes, NY, 2018.
Argyros, I.K., George, S., Thapa, N., Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications, Vol. II, Nova Publishes, NY, 2018.
Argyros, I.K., Hilout, S., Weaker conditions for the convergence of Newtons method, J. Complexity, 28(2012), no. 3, 364-387.
Cordero, A., Hueso, J. L., Mart´ınez, E., Torregrosa, J.R., A modified Newton-Jarratt’s composition, Numer. Algorithms, 55(2010), 87-99.
Grau-Sa´nchez, M., Noguera, M., Amat, S., On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. Appl. Math., 237(2013), 363-372.
Lotfi, T., Bakhtiari, P., Cordero, A., Mahdiani, K., Torregrosa, J.R., Some new efficient multipoint iterative methods for solving nonlinear systems of equations, Int. J. Comput. Math., 92(2015), 1921-1934.
Madhu, K., Babajee, D.K.R., Jayaraman, J., An improvement to double-step Newton method and its multi-step version for solving system of nonlinear equations and its applications, Numer. Algorithms, 74(2017), 593-607.
Magrenan, A.A., Argyros, I.K., Two-step Newton methods, J. Complexity, 30(2014), no. 4, 533-553.
Magrenan, A.A., Argyros, I.K., Improved convergence analysis for Newton-like methods, Numer. Algorithms, 71(2016), no. 4, 811-826.
Magrenan, A.A., Cordero, A., Gutierrez, J.M., Torregrosa, J.R., Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane, Math. Comput. Simulation, 105(2014), 49-61.
Narang, M., Bhatia, S., Alshomrani, A.S., Kanwar, V., General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations, J. Comput. Appl. Math., 352(2019), 23-39.
Osrowski, A.M., Solution of Equations and Systems of Equations, Academic Press, New York, 1960.
Potra, F.A., Ptak, V., Nondiscrete Induction and Iterative Processes, vol. 103, Pitman Advanced Publishing Program, 1984.
Rheinboldt, W.C., An adaptive continuation process for solving systems of nonlinear equations, in: Mathematical models and numerical methods (A.N. Tikhonov et al. eds.), Banach Center, Warsaw Poland, pub. 3, 1977, 129-142.
Sharma, J.R., Argyros, I.K., Kumar, S., Ball convergence of an efficient eighth order iterative method under weak conditions, Mathematics, 6(2018), no. 11, 260, https://doi.org/10.3390/math6110260.
Sharma, J.R., Arora, H., Improved Newton-like methods for solving systems of nonlinear equations, SeMA J., 74(2017), no. 2, 147-163.
Sharma, J.R., Arora, H., A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algorithms, 67(2014), 917-933.
Traub, J.F., Iterative Methods for the Solution of Equations, AMS Chelsea Publishing, 1982.
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