We study the problem of local convergence of the accelerated Newton method for the solution of nonlinear functional equations under generalized Lipschitz conditions for the first- and second-order Fréchet derivatives. We show that the accelerated method is characterized by the quadratic order of convergence and compare it with the classical Newton method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Ya. Bartish and S. M. Shakhno, “On the Newton method with accelerated convergence,” Vestn. Kiev. Univ., Modelir. Optimiz. Slozhn. Sistem, No. 6, 62–66 (1987).
M. Ya. Bartish and S. M. Shakhno, On a Class of Newton-Type Iterative Methods [in Russian], Deposited at UkrNIINTI 17.09.87, No. 2580-Uk87, Lviv (1987).
L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1984).
J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic, New York (1970).
S. M. Shakhno, “Convergence of inexact difference methods under the generalized Lipschitz conditions,” Mat. Metody Fiz.-Mekh. Polya, 52, No. 3, 30–40 (2009); English translation: J. Math. Sci., 171, No. 4, 453–465 (2010).
S. M. Shakhno, “Steffensen method under generalized Lipschitz conditions for the first-order divided differences,” Mat. Stud., 31, No. 1, 90–95 (2009).
S. M. Shakhno, Investigation of Convergence of the Accelerated Newton Method [in Russian], Deposited at UkrNIINTI 17.09.87, No. 2579-Uk87, Lviv (1987).
J. Chen and W. Li, “Convergence behavior of inexact Newton methods under weak Lipschitz condition,” J. Comput. Appl. Math., 191, No. 1, 143–164 (2006).
S. M. Shakhno, “On the secant method under generalized Lipschitz conditions for the divided difference operator,” Proc. Appl. Math. Mech., 7, No. 1, 2060083–2060084 (2007).
X. Wang, “Convergence of Newton’s method and uniqueness of the solution of equations in Banach space,” IMA, J. Numer. Anal., 20, No. 1, 123–134 (2000).
W. Werner, “Newton-like methods for the computation of fixed points,” Comput. & Math. Appl., 10, No. 1, 77–86 (1984).
W. Werner, “Über ein Verfahren der Ordnung \( 1+\sqrt{2} \) zur Nullstellenbestimmung,“ Numer. Math., 32, No. 3, 333–342 (1979).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 57, No. 1, pp. 18–25, January–March, 2014.
Rights and permissions
About this article
Cite this article
Shakhno, S.М. On Convergence of the Accelerated Newton Method Under Generalized Lipschitz Conditions. J Math Sci 212, 16–26 (2016). https://doi.org/10.1007/s10958-015-2645-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-015-2645-5