Abstract
We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. K. Argyros: Polynomial operator equations in abstract spaces and applications. St. Lucie/CRC/Lewis Publ. Mathematics series, 1998, Boca Raton, Florida, U.S.A.
I. K. Argyros: On the Newton-Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169 (2004), 315–332.
I. K. Argyros: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space. J. Math. Anal. Appl. 298 (2004), 374–397.
I. K. Argyros: New sufficient convergence conditions for the Secant method. Chechoslovak Math. J. 55 (2005), 175–187.
I. K. Argyros: Convergence and Applications of Newton-Type Iterations. Springer-Verlag Publ., New-York, 2008.
I. K. Argyros and S. Hilout: Efficient Methods for Solving Equations and Variational Inequalities. Polimetrica Publisher, 2009.
W. E. Bosarge and P. L. Falb: A multipoint method of third order. J. Optimiz. Th. Appl. 4 (1969), 156–166.
S. Chandrasekhar: Radiative Transfer. Dover Publ., New-York, 1960.
J. E. Dennis: Toward a unified convergence theory for Newton-like methods. In Nonlinear Functional Analysis and Applications (L.B. Rall, ed.), Academic Press, New York (1971), 425–472.
M. A. Hernández, M. J. Rubio and J.A. Ezquerro: Solving a special case of conservative problems by Secant-like method. Appl. Math. Cmput. 169 (2005), 926–942.
M. A. Hernández, M. J. Rubio and J.A. Ezquerro: Secant-like methods for solving nonlinear integral equations of the Hammerstein type. J. Comput. Appl. Math. 115 (2000), 245–254.
Z. Huang: A note of Kantorovich theorem for Newton iteration. J. Comput. Appl. Math. 47 (1993), 211–217.
L. V. Kantorovich and G. P. Akilov: Functional Analysis. Pergamon Press, Oxford, 1982.
P. Laasonen: Ein überquadratisch konvergenter iterativer Algorithmus. Ann. Acad. Sci. Fenn. Ser I 450 (1969), 1–10.
J. M. Ortega and W.C. Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.
F. A. Potra: Sharp error bounds for a class of Newton-like methods. Libertas Mathematica 5 (1985), 71–84.
J. W. Schmidt: Untere Fehlerschranken fur Regula-Falsi Verfahren. Period. Hungar. 9 (1978), 241–247.
T. Yamamoto: A convergence theorem for Newton-like methods in Banach spaces. Numer. Math. 51 (1987), 545–557.
M. A. Wolfe: Extended iterative methods for the solution of operator equations. Numer. Math. 31 (1978), 153–174.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Argyros, I.K., Hilout, S. Convergence conditions for secant-type methods. Czech Math J 60, 253–272 (2010). https://doi.org/10.1007/s10587-010-0014-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-010-0014-6