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
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.
nonlinear equations, Newton-type iterations, order of convergence, optimal order.
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Computational Mathematics and Mathematical Physics, 2017, 57:7, 1090–1100
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.
\jour Comput. Math. Math. Phys.
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Zhanlav T., Chuluunbaatar O., Ulziibayar V., “Generating Function Method For Constructing New Iterations”, Appl. Math. Comput., 315 (2017), 414–423
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
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