Extended local and semilocal convergence for interpolatory iterative methods for nonlinear equations
Tóm tắt
Two iterative methods free of the Fréchet derivative of a nonlinear operator are considered. A local and a semilocal convergence analysis of the methods is carried out under Lipschitz conditions for the first-order divided differences and Hölder conditions for the second-order divided differences. The dependence of the convergence orders of methods on the Hölder constant and uniqueness ball are established. In this article we use our new idea of the restricted convergence region and present an improved convergence analysis of these methods. A numerical example, which demonstrates advantages of our approach to study convergence of iterative methods for solving nonlinear equations, is given.
Tài liệu tham khảo
Amat, S., Bermúdez, C., Hernández-Verón, M.A., Martínez, E.: On an efficient k-step iterative method for nonlinear equations. J. Comput. Appl. Math. 302, 258–271 (2016)
Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Lett. 25(12), 2209–2217 (2012)
Argyros, I.K.: A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations. J. Math. Anal. Appl. 332, 97–108 (2007)
Argyros, I.K., Magreñán, A.A.: Iterative Methods and Their Dynamics with Applications: A Contemporary Study. CRC Press, Boca Raton (2017)
Argyros, I.K., Shakhno, S., Yarmola, H.: Extending the Convergence region of Methods of Linear Interpolation for the Solution of Nonlinear Equations. Symmetry. (2020). https://doi.org/10.3390/sym12071093
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)
Kurchatov, V.A.: On a method of linear interpolation for the solution of functional equations. Dokl. Akad. Nauk SSSR 198(3), 524–526 (1971). (in Russian)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Potra, F.A.: On an iterative algorithm of order 1.839... for solving nonlinear operator equations. Numer. Funct. Anal. Optim. 7(1), 75–106 (1985)
Shakhno, S.: Combined Newton–Kurchatov method under the generalized Lipschitz conditions for the derivatives and divided differences. J. Comput. Appl. Math. (Kyiv) 2(119), 78–89 (2015)
Shakhno, S.M.: Method of linear interpolation of Kurchatov under generalized Lipschitz conditions for divided differences of first and second order. Visnyk Lviv Univ Ser Mech Math. 77, 235–242 (2012). (in Ukrainian)
Shakhno, S.M.: On the difference method with quadratic convergence for solving nonlinear operator equations. Matematychni Studii. 26, 105–110 (2006). (in Ukrainian)
Shakhno, S.M., Babjak, A.-V.I., Yarmola, H.P.: Combined Newton–Potra method for solving nonlinear operator equations. J. Comput. Appl. Math. (Kyiv) 3(120), 170–178 (2015). (in Ukrainian)
Shakhno, S.M., Makukh, O.M.: About iterative methods in conditions of Holder continuity of the method divided differences of the second order. Math. Methods Physicomech. Field. 49(2), 90–98 (2006). (in Ukrainian)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall Inc, New York (1964)
Ul’m, S.: Algorithms of the generalized Steffensen method. Izv. Akad. Nauk ESSR, Ser. Fiz.-Mat. 14, 433–443 (1965). (in Russian)
Ul’m, S.: On generalized divided differences I. Izv. Akad. Nauk ESSR, Ser. Fiz.-Mat 16, 13–26 (1967). (in Russian)