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Open AccessArticle

Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations

1
Department of Mathematics, Cameron University, Lawton, OK 73505, USA
2
Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universitetska Str. 1, 79000 Lviv, Ukraine
3
Department of Computational Mathematics, Ivan Franko National University of Lviv, Universitetska Str. 1, 79000 Lviv, Ukraine
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(7), 1093; https://doi.org/10.3390/sym12071093
Received: 26 May 2020 / Revised: 27 June 2020 / Accepted: 29 June 2020 / Published: 1 July 2020
(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)
Solving equations in abstract spaces is important since many problems from diverse disciplines require it. The solutions of these equations cannot be obtained in a form closed. That difficulty forces us to develop ever improving iterative methods. In this paper we improve the applicability of such methods. Our technique is very general and can be used to expand the applicability of other methods. We use two methods of linear interpolation namely the Secant as well as the Kurchatov method. The investigation of Kurchatov’s method is done under rather strict conditions. In this work, using the majorant principle of Kantorovich and our new idea of the restricted convergence domain, we present an improved semilocal convergence of these methods. We determine the quadratical order of convergence of the Kurchatov method and order 1 + 5 2 for the Secant method. We find improved a priori and a posteriori estimations of the method’s error.
Keywords: nonlinear equation; iterative process; convergence order; secant method; Kurchatov method; Banach space; divided difference; local; semi-local convergence nonlinear equation; iterative process; convergence order; secant method; Kurchatov method; Banach space; divided difference; local; semi-local convergence
MDPI and ACS Style

K. Argyros, I.; Shakhno, S.; Yarmola, H. Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations. Symmetry 2020, 12, 1093.

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