Extending the applicability of Newton's and Secant methods under regular smoothness

Ioannis K. Argyros; Santhosh George; Shobha M. Erappa

doi:10.5269/bspm.42132

Ioannis K. Argyros Cameron University https://orcid.org/0000-0002-9189-9298
Santhosh George National Institute of Technology Karnataka https://orcid.org/0000-0002-3530-5539
Shobha M. Erappa Manipal Institute of Technology https://orcid.org/0000-0002-6523-0873

Résumé

The concept of regular smoothness has been shown to be an appropriate and powerfull tool for the convergence of iterative procedures converging to a locally unique solution of an operator equation in a Banach space setting. Motivated by earlier works, and optimization considerations, we present a tighter semi-local convergence analysis using our new idea of restricted convergence domains. Numerical examples complete this study.

Téléchargements

Les données sur le téléchargement ne sont pas encore disponible.

Bibliographies de l'auteur

Ioannis K. Argyros, Cameron University

Professor,
Department of Mathematics Sciences

Santhosh George, National Institute of Technology Karnataka

Professor,

Department of Mathematical and Computational Sciences

Shobha M. Erappa, Manipal Institute of Technology

Department of Mathematics

Références

Amat, S., Busquier,S., Plaza, S., Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Appl. 366, 1, (2010), 24-32.

Argyros I. K., A unifying local–semilocal convergence analysis and applications for two–point Newton–like methods in Banach space, J. Math. Anal. and Appl., 298 (2004), 374–397.

Argyros I. K., Convergence and applications of Newton–type iterations, Springer, New York, 2008.

Argyros, I. K., Hilout, S., Weaker conditions for the convergence of Newton’s method, Journal of Complexity, 28, 3, (2012), 364-387.

Cianciaruso, F., A further journey in the ”Terra Incognitta” of the Newton–Kantorovich method, Nonlinear Funct. Anal. Appl., 15, (2010) n0.2, 173-183.

Galperin, A., Secant method with regularly continuous divided differences, J. Comput. Appl. Math., 193 (2006), no. 2, 574–595.

Galperin, A., Waksman, Z., Regular smoothness and Newton’s method, Numer. Funct. Anal. Optim., 15 (1994), 813–858.

Hernanadez, M. A., Rubio, M. J., Ezquerro, J. A., Secant–like methods for solving integral equations of the Hammerstein type, J. Comput. Appl. Math., 115 (2001), 245–254.

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

Kornstaedt, H. J., Funktionallongleichungen und iterations verfahren, Aequationes Math., 13 (1975), 21–45.

Magren˜an, A. A., Different anomalies in a Jarratt family of iterative root-finding methods, Appl.Math.Comput.233, (2014), 29-38.

Magrenan, A. A., A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, (2014), 215-224.

Potra, F. A., An application of the induction method of V. Ptak to the study of regula falsi, Aplikace Matematiky, 26 (1981), 111–120.

Potra, F. A., An error analysis for the secant method, Numer. Math., 38 (1981/82), 427–445.

Schmidt, J. W., Uberlinear konvergente Mehrschrittverfahren vom Regula falsi– und Newton–Typ, (German), Z. Angew. Math. Mech., 53 (1973), 103–114.

Schmidt, J. W., Eine Ubertragung der Regula Falsi auf Gleichungen in Banachrau men, I, Nichtlineare Gleichungssysteme, (German), Z. Angew. Math. Mech., 43 (1963), 1–8.

Schmidt, J. W., Eine Ubertragung der Regula Falsi auf Gleichungen in Banachrau men, II, Nichtlineare Gleichungssysteme, (German), Z. Angew. Math. Mech., 43 (1963), 97–110.

Sergeev, A. S., The method of chords, (Russian), Sibirsk. Mat. Z., 2 (1961), 282–289.