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Mathematics 2019, 7(3), 299; https://doi.org/10.3390/math7030299

Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications

1
Department of Mathematics Sciences Lawton, Cameron University, Lawton, OK 73505, USA
2
Departamento de Matemáticas y Computación, Universidad de La Rioja, 26006 Logroño, Spain
3
Departamento de Matemática Aplicada, Universidad Politècnica de València, 46022 València, Spain
4
Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, 26006 Logroño; Spain
*
Author to whom correspondence should be addressed.
Received: 3 March 2019 / Revised: 20 March 2019 / Accepted: 21 March 2019 / Published: 24 March 2019
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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Abstract

The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of restricted convergence domains, we extend the applicability of Newton’s method as follows: The convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. These advantages are obtained using the same information as before, since new Lipschitz constant are tighter and special cases of the ones used before. Numerical examples and applications are used to test favorable the theoretical results to earlier ones. View Full-Text
Keywords: Banach space; Newton’s method; semi-local convergence; Kantorovich hypothesis Banach space; Newton’s method; semi-local convergence; Kantorovich hypothesis
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Argyros, I.K.; Magreñán, Á.A.; Orcos, L.; Sarría, Í. Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications. Mathematics 2019, 7, 299.

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