Extending the Applicability of the Super-Halley-like Method Using ω-continuous derivatives and Restricted Convergence Domains

• Autorzy: Ioannis K. Argyros , Santhosh George
• Języki publikacji: EN

Abstrakty

EN

We present a local convergence analysis of the super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier studies was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the super-Halley-like method by using hypotheses up to the second derivative. We also provide: a computable error on the distances involved and a uniqueness result based on Lipschitz constants. Numerical examples are also presented in this study. (original abstract)

Słowa kluczowe

EN

Mathematics; Mathematical functions; Mathematical statistics; Convergence

PL

Matematyka; Funkcje matematyczne; Statystyka matematyczna; Konwergencja

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2019
• Tom: 33
• Strony: 21--40

Twórcy

autor

Ioannis K. Argyros

• Cameron University Department of Mathematical Sciences

autor

Santhosh George

• National Institute of Technology Karnataka Department of Mathematical and Computational Sciences

Bibliografia

• Argyros I.K., On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 169 (2004), 315-332.
• Argyros I.K., Ezquerro J.A., Gutiérrez J.M., Hernández M.A., Hilout S., On the semilocal convergence of efficient Chebyshev-Secant-type methods, J. Comput. Appl. Math. 235 (2011), 3195-3206.
• Argyros I.K., Ren H., Efficient Steffensen-type algorithms for solving nonlinear equations, Int. J. Comput. Math. 90 (2013), 691-704.
• Argyros I.K., Computational Theory of Iterative Methods, Studies in Computational Mathematics, 15, Elsevier B.V., New York, 2007.
• Ezquerro J.A., Hernández M.A., An optimization of Chebyshev's method, J. Complexity 25 (2009), 343-361.
• Ezquerro J.A., Grau A., Grau-Sánchez M., Hernández M.A., Construction of derivative-free iterative methods from Chebyshev's method, Anal. Appl. (Singap.) 11 (2013), 1350009, 16 pp.
• Ezquerro J.A., Gutiérrez J.M., Hernández M.A., Salanova M.A., Chebyshev-like methods and quadratic equations, Rev. Anal. Numér. Théor. Approx. 28 (1999), 23-35.
• Grau M., Díaz-Barrero J.L., An improvement of the Euler-Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006), 1-7.
• Grau-Sánchez M., Gutiérrez J.M., Some variants of the Chebyshev-Halley family of methods with fifth order of convergence, Int. J. Comput. Math. 87 (2010), 818-833.
• Hueso J.L., Martinez E., Teruel C., Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems, J. Comput. Appl. Math. 275 (2015), 412-420.
• Magreñán Á.A., Estudio de la dinámica del método de Newton amortiguado, PhD Thesis, Universidad de La Rioja, Servicio de Publicaciones, Logroño, 2013. Available at http://dialnet.unirioja.es/servlet/tesis?codigo=38821
• Magreñán Á.A., Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput. 233 (2014), 29-38.
• Magreñán Á.A., A new tool to study real dynamics: the convergence plane, Appl. Math. Comput. 248 (2014), 215-224.
• Prashanth M., Mosta S.S., Gupta D.K., Semi-local convergence of the Supper-Halley's method under w-continuous second derivative in Banach space. Submitted.
• Rheinboldt W.C., An adaptive continuation process for solving systems of nonlinear equations, in: Tikhonov A.N., et al. (eds.), Mathematical Models and Numerical Methods, Banach Center Publ., 3, PWN, Warsaw, 1978, pp. 129-142.
• Traub J.F., Iterative Methods for the Solution of Equations, Prentice-Hall Series in Automatic Computation, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1964.

Identyfikatory

DOI

10.2478/amsil-2018-0008