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Zh. Vychisl. Mat. Mat. Fiz., 2017, Volume 57, Number 7, Pages 1093–1102 (Mi zvmmf10583)  

This article is cited in 2 scientific papers (total in 2 papers)

Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations

T. Zhanlava, V. Ulziibayarab, O. Chuluunbaatarca

a Institute of Mathematics, National University of Mongolia, Ulan-Bator, Mongolia
b Mongolian University of Science and Technology, Ulan-Bator, Mongolia
c Joint Institute for Nuclear Research, Dubna, Moscow oblast, Russia

Abstract: Necessary and sufficient conditions under which two- and three-point iterative methods have the order of convergence $p$ ($2\leqslant p\leqslant 8$) are formulated for the first time. These conditions can be effectively used to prove the convergence of iterative methods. In particular, the order of convergence of some known optimal methods is verified using the proposed sufficient convergence tests. The optimal set of parameters making it possible to increase the order of convergence is found. It is shown that the parameters of the known iterative methods with the optimal order of convergence have the same asymptotic behavior. The simplicity of choosing the parameters of the proposed methods is an advantage over the other known methods.

Key words: nonlinear equations, Newton-type iterations, order of convergence, optimal order.

Funding Agency Grant Number
Фонд науки и технологии Монголии SST_007/2015
Joint Institute for Nuclear Research 05-6-1119-2014/2019

DOI: https://doi.org/10.7868/S0044466917070146

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English version:
Computational Mathematics and Mathematical Physics, 2017, 57:7, 1090–1100

Bibliographic databases:

UDC: 519.61
Received: 29.04.2016

Citation: T. Zhanlav, V. Ulziibayar, O. Chuluunbaatar, “Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations”, Zh. Vychisl. Mat. Mat. Fiz., 57:7 (2017), 1093–1102; Comput. Math. Math. Phys., 57:7 (2017), 1090–1100

Citation in format AMSBIB
\by T.~Zhanlav, V.~Ulziibayar, O.~Chuluunbaatar
\paper Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2017
\vol 57
\issue 7
\pages 1093--1102
\jour Comput. Math. Math. Phys.
\yr 2017
\vol 57
\issue 7
\pages 1090--1100

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Zhanlav T., Chuluunbaatar O., Ulziibayar V., “Generating Function Method For Constructing New Iterations”, Appl. Math. Comput., 315 (2017), 414–423  crossref  mathscinet  isi
    2. T. Zhanlav, O. Chuluunbaatar, V. Ulziibayar, “Generating function approach to the derivation of higher-order iterative methods for solving nonlinear equations”, Mathematical Modeling and Computational Physics 2017 (MMCP 2017), EPJ Web Conf., 173, eds. G. Adam, J. Busa, M. Hnatic, D. Podgainy, EDP Sciences, 2018, 03024  crossref  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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