Ball convergence of a novel bi-parametric iterative scheme for solving equations
Downloads
DOI:
https://doi.org/10.26637/MJM0803/0087Abstract
The aim of this article is to establish a ball convergence result for a bi-parametric iterative scheme for solving equations involving Banach space valued operators. In contrast to earlier approaches in the less general setting of the k-dimensional Euclidean space where hypotheses on the seventh derivative are used, we only use hypotheses on the first derivative. Hence, we extend the applicability of the method. Moreover, the radius of convergence as well as error bounds on the distances are given based on Lipschitz-type functions. Numerical examples are given to test our conditions. These examples show that earlier convergence conditions are not satisfied but ours are satisfied.
Keywords:
Bi-parametric iterative scheme, Banach space, Ball convergence, Lipschitz-type conditions, Frechet SpacesMathematics Subject Classification:
Mathematics- Pages: 1228-1233
- Date Published: 01-07-2020
- Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)
Argyros, I. K., George, S., Magreñán, A. A., Local convergence for multi-point-parametric Chebyshev-Halleytype methods of high convergence order, Journal of Computational and Applied Mathematics, 282(2015), $215-$ 224.
Argyros, I. K., George, S., Thapa, N., Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications, Volume-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, Volume-II, Nova Publishes, NY, 2018.
Argyros, I. K, Kansal, M., Kanwar, K., Local convergence for multi-point methods using only the first derivative, SeMA, DOI 10.1007/s40324-016-0075-z.
Bahl. A, Cordero. A, Sharma. R, Torregrosa.J, Analysing complex dynamics of a novel bi-parametric sixth order iterative scheme for solving nonlinear systems, Appl. Math. Comput., (To appear).
Candela, V., Marquina, A. , Recurrence relations for rational cubic methods I: The Halley method, Computing, $44(2)(1990), 169-184$.
Candela, V., Marquina, A., Recurrence relations for rational cubic methods II: The Halley method, Computing, 45(4)(1990), 355-367.
Chen, J., Argyros, I. K., Agarwal, R. P., Majorizing functions and two-point Newton-type methods, J. Comput. Appl. Math., 224(5)(2010), 1473-1484.
Ezquerro, J.A., González,D., Hernández, M.A., On the local convergence of Newton's method under generalized conditions of Kantorovich, Applied Mathematics Letters, 26(2013), 566-570.
Gutierrez, J. M., Hernández, M. A., An acceleration of Newtons method: super-Halley method, Appl. Math. Comput., 117(2001), 223-239.
Hernandez, M. A., Romero, N., On a characterization of some Newton-like method of $r$-order at least three, $J$. Comput. Appl. Math., 183(2005), 53-66.
Hernandez, M. A. H., Martinez, E., On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions, Numer. Algor., $70(2)(2015), 377-392$.
Jarratt, P., Some fourth order multipoint iterative methods for solving equations, Math. Comput., 20(1966), 434 437.
Kantorovich, L.V., Akilov, G.P., Functional Analysis, Pergamon Press, Oxford, 1982.
Kumar. A, Gupta. D. K., Local convergence of Super Halley's method under weaker conditions on Fréchet derivative in Banach spaces, The Journal of Analysis, (2017), Doi.org/10.1007/s41478-017-0034-9.
Madhu, K., Sixth order Newton-type method for solving system of nonlinear equations and its applications, Applied Mathematics E- Notes, 7(2017), 221-230.
Magreñán, Á. A., Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput., 233(2014), 29-38.
Magreñán, Á. A., A new tool to study real dynamics: The convergence plane, Applied Mathematics and Computation, 248(2014), 215-224.
${ }^{[19]}$ Parida,P. K., Gupta, D. K., Recurrence relations for a Newton-like method in Banach space, J. Comput. Appl. Math., 206(2007), 873-887.
Rheinboldt, W.C., An adaptive continuation process for solving systems of nonlinear equations, Polish Academy of Science, Banach Ctr. Publ., 3(1)(1978), 129-142.
Traub, J. F., Iterative Methods for Solution of Equations, Printice-Hall, Inc., Englewood Cliffs, N. J., 1964.
Wang, K., Kou, J., Gu, C., Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algor., 57(2011), 441-456.
- NA
Similar Articles
- M.J. Abisha, K. Rubin Mary, k-Lehmer three mean labeling of some graphs , Malaya Journal of Matematik: Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)
You may also start an advanced similarity search for this article.
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 MJM
This work is licensed under a Creative Commons Attribution 4.0 International License.
