On the convergence of Kurchatov-type methods using recurrent functions for solving equations

I. K. Argyros; S. Shakhno; H. Yarmola

doi:10.30970/ms.58.1.103-112

I. K. Argyros Department of Mathematical Sciences, Cameron University, Lawton, USA
S. Shakhno Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Lviv, Ukraine
H. Yarmola Department of Computational Mathematics, Ivan Franko National University of Lviv, Lviv, Ukraine

DOI: https://doi.org/10.30970/ms.58.1.103-112

Keywords: nonlinear equation; Kurchatov's method; Banach space; divided difference; local and semi-local convergence; two-step method

Abstract

We study a local and semi-local convergence of Kurchatov's method and its two-step modification for solving nonlinear equations under the classical Lipschitz conditions for the first-order divided differences. To develop a convergence analysis we use the approach of restricted convergence regions in a combination to our technique of recurrent functions. The semi-local convergence is based on the majorizing scalar sequences. Also, the results of the numerical experiment are given.

Author Biographies

I. K. Argyros , Department of Mathematical Sciences, Cameron University, Lawton, USA

Department of Mathematical Sciences,
Cameron University, Lawton, USA

S. Shakhno, Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Lviv, Ukraine

Department of Theory of Optimal Processes,
Ivan Franko National University of Lviv, Lviv, Ukraine

H. Yarmola, Department of Computational Mathematics, Ivan Franko National University of Lviv, Lviv, Ukraine

Department of Computational Mathematics,
Ivan Franko National University of Lviv, Lviv, Ukraine

References

I.K. Argyros, A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations, J. Math. Anal. Appl., 332 (2007), 97–108.

I.K. Argyros, S. George, On a two-step Kurchatov-type method in Banach space, Mediterr. J. Math., 16, Article number: 21 (2019). https://doi.org/10.1007/s00009-018-1285-7

I.K. Argyros, A.A. Magrenan, Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press, Boca Raton Florida, USA, 2017.

I.K. Argyros, S. Shakhno, Extended two-step-Kurchatov method for solving Banach space valued nondifferentiable equations, Int. J. Appl. Comput. Math., 6, Article number: 32, (2020). https://doi.org/10.1007/s40819-020-0784-y

J.E. Dennis, Jr. Schnabel, B. Robert, Numerical methods for unconstrained optimization and nonlinear equations, Englewood Cliffs, Prentice-Hall, 1983.

L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.

V.A. Kurchatov, On a method of linear interpolation for the solution of functional equations, Dokl. Akad. Nauk. SSSR. 198 (1971), №3, 524–526 (in Russian); translation in Soviet. Math. Dokl. 12 (1971), 835–838.

S.M. Shakhno, Kurchatov method of linear interpolation under generalized Lipschitz conditions for divided differences of first and second order, Visnyk Lviv Univ. Ser. Mech. Math., 77 (2012), 235–242. (in Ukrainian)

S.M. Shakhno, On a Kurchatov’s method of linear interpolation for solving nonlinear equations, PAMM: Proceedings in Applied Mathematics and Mechanics, 4 (2004), №1, 650–651.

S.M. Shakhno, On the difference method with quadratic convergence for solving nonlinear operator equations,Mat. Stud., 26 (2006), №1, 105–110. (in Ukrainian)

S.M. Shakhno, Nonlinear majorants for investigation of methods of linear interpolation for the solution of nonlinear equations, In: ECCOMAS 2004 - European congress on computational methods in applied sciences and engineering; P. Neittaanmaki,T. Rossi, K. Majava, and O. Pironneau (eds.) O. Nevanlinna and R. Rannacher (assoc. eds.) Jyvaskyla, 24-28 July 2004.

S. Ul’m, On generalized divided differences I, II, Izv. Akad. Nauk. ESSR, Ser. Fiz.-Mat., 16 (1967), 13–26, 146–156. (in Russian)