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http://dx.doi.org/10.4134/BKMS.b180429

IMPROVED LOCAL CONVERGENCE ANALYSIS FOR A THREE POINT METHOD OF CONVERGENCE ORDER 1.839…  

Argyros, Ioannis K. (Department of Mathematical Sciences Cameron University)
Cho, Yeol Je (Department of Mathematics Education Gyeongsang National University)
George, Santhosh (Department of Mathematical and Computational Sciences National Institute of Technology Karnataka)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.3, 2019 , pp. 621-629 More about this Journal
Abstract
In this paper, we present a local convergence analysis of a three point method with convergence order $1.839{\ldots}$ for approximating a locally unique solution of a nonlinear operator equation in setting of Banach spaces. Using weaker hypotheses than in earlier studies, we obtain: larger radius of convergence and more precise error estimates on the distances involved. Finally, numerical examples are used to show the advantages of the main results over earlier results.
Keywords
Banach space; three point method; divided difference of order one-two; radius of convergence; local convergence;
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1 S. Amat, S. Busquier, and J. M. Gutierrez, Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 157 (2003), no. 1, 197-205.   DOI
2 I. K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004), no. 2, 374-397.   DOI
3 I. K. Argyros, Computational Theory of Iterative Methods, Elsevier Publ. Co. New York, USA, 2007.
4 I. K. Argyros, A semilocal convergence analysis for directional Newton methods, Math. Comp. 80 (2011), no. 273, 327-343.   DOI
5 I. K. Argyros, Y. J. Cho, and S. George, On the "terra incognita" for the Newton-Kantrovich method with applications, J. Korean Math. Soc. 51 (2014), no. 2, 251-266.   DOI
6 I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical methods for equations and its applications, CRC Press, Boca Raton, FL, 2012.
7 I. K. Argyros, Y. J. Cho, and H. Ren, Convergence of Halley's method for operators with the bounded second Frechet-derivative in Banach spaces, J. Inequal. Appl. 2013 (2013), 260, 12 pp.   DOI
8 I. K. Argyros and S. Hilout, Weaker conditions for the convergence of Newton's method, J. Complexity 28 (2012), no. 3, 364-387.   DOI
9 V. Candela and A. Marquina, Recurrence relations for rational cubic methods: I. The Halley method, Computing 44 (1990), no. 2, 169-184.   DOI
10 R. Behl, A. Cordero, S. S. Motsa, and J. R. Torregrosa, On developing fourth-order optimal families of methods for multiple roots and their dynamics, Appl. Math. Comput. 265 (2015), 520-532.   DOI
11 V. Candela and A. Marquina, Recurrence relations for rational cubic methods. II. The Chebyshev method, Computing 45 (1990), no. 4, 355-367.   DOI
12 C. Chun, P. Stanica, and B. Neta, Third-order family of methods in Banach spaces, Comput. Math. Appl. 61 (2011), no. 6, 1665-1675.   DOI
13 M. A. Hernandez and M. A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math. 126 (2000), no. 1-2, 131-143.   DOI
14 J. M. Gutierrez and M. A. Hernandez, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997), no. 1-2, 171-183.   DOI
15 J. M. Gutierrez and M. A. Hernandez, Recurrence relations for the super-Halley method, Comput. Math. Appl. 36 (1998), no. 7, 1-8.
16 M. A. Hernandez, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001), 433-455.   DOI
17 L. V. Kantorovich and G. P. Akilov, Functional Analysis, translated from the Russian by Howard L. Silcock, second edition, Pergamon Press, Oxford, 1982.
18 I. K. Argyros and S. Hilout, Computational Methods in Nonlinear Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
19 A. Magrenan, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput. 233 (2014), 29-38.   DOI
20 A. Magrenan, A new tool to study real dynamics: the convergence plane, Appl. Math. Comput. 248 (2014), 215-224.   DOI
21 A. Magrenan and I. K. Argyros, A contemporary study of iterative methods, Academic Press, London, 2018.
22 J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
23 J. R. Sharma and H. Arora, Efficient Jarratt-like methods for solving systems of non-linear equations, Calcolo 51 (2014), no. 1, 193-210.   DOI
24 P. K. Parida and D. K. Gupta, Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces, J. Math. Anal. Appl. 345 (2008), no. 1, 350-361.   DOI
25 M. S. Petkovic, B. Neta, L. Petkovic, and J. Dzunic, Multipoint Methods for Solving Nonlinear Equations, Elsevier/Academic Press, Amsterdam, 2013.
26 F. A. Potra, On an iterative algorithm of order 1.839...for solving nonlinear operator equations, Numer. Funct. Anal. Optim. 7 (1984/85), no. 1, 75-106.   DOI