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Section
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Mathematics
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Title
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One specification of Steffensen's method for solving nonlinear operator equations
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Author(-s)
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Yumanova I.F.a
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Affiliations
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Ural Federal Universitya
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Abstract
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We consider an analogue of Steffensen's method for solving nonlinear operator equations. The proposed method is a two-step iterative process. We study the convergence of the proposed method, prove the uniqueness of the solution and find the order of convergence. The proposed method uses no derivative operators. The convergence order is greater than that in Newton's method and some generalizations of the method of chords and Aitken-Steffensen's method. The method is applied to some test systems of nonlinear equations and the problem of curves intersection which are defined implicitly as solutions of differential equations. Numerical results are compared with the results obtained by Newton's method, the modified Newton method, and modifications of Wegstein's and Aitken's methods which were proposed by the author in previous works.
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Keywords
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nonlinear operator equation, Steffensen's method, Newton's method, problem of the intersection curves
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UDC
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519.615.5, 519.642.6
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MSC
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45G10, 65J15
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DOI
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10.20537/vm160411
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Received
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30 October 2016
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Language
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Russian
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Citation
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Yumanova I.F. One specification of Steffensen's method for solving nonlinear operator equations, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 579-590.
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References
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- Ostrowski A.M. Solution of equations and systems of equations, New York: Academic Press, 1960, 202 p. Translated under the title Reshenie uravnenii i sistem uravnenii, Moscow: Inostr. Lit., 1963, 220 p.
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- Vainberg M.M. Variatsionnye metody issledovaniya nelineinykh operatorov (Variational methods for the study of nonlinear operators), Moscow: Gos. Izd. Tekh. Teor. Lit., 1956, 345 p.
- Roose A., Kulla V., Lomp M., Meressoo T. Nabor testovykh sistem nelineinykh uravnenii. Izd. 2 (A set of test systems of nonlinear equations. Ed. 2), Tallinn: Valgus, 1989, 132 p.
- Yumanova I.F. On application of the Wegstein method to nonlinear systems, Sovremennye problemy matematiki. Tezisy mezhdunarodnoi (44-oi Vserossiiskoi) molodezhnoi shkoly-konferentsii (Actual problems of mathematics. Proceedings of International (44th All-Russian) Youth School-Conference), Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 2013, pp. 166-169 (in Russian).
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