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On Newton’s Method for Solving Nonlinear Equations and Function Splitting

Published online by Cambridge University Press:  28 May 2015

Ioannis K. Argyros  and
Saïd Hilout
Ioannis K. Argyros
Affiliation:
Cameron University, Department of Mathematics Sciences, Lawton, OK 73505, USA
Saïd Hilout
Affiliation:
Université de Poitiers, Laboratoire de Mathématiques et Applications, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179, 86962 Futuroscope Chasseneuil Cedex, France
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Abstract

We provided in [14] and [15] a semilocal convergence analysis for Newton’s method on a Banach space setting, by splitting the given operator. In this study, we improve the error bounds, order of convergence, and simplify the sufficient convergence conditions. Our results compare favorably with the Newton-Kantorovich theorem for solving equations.

Keywords

65H9965H1065G9949M1547J2047H0490C3090C33Newton’s methodBanach spacemajorizing sequencesemilocal convergencesplitting of an operator
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 1 , February 2011 , pp. 53 - 67
Copyright
Copyright © Global Science Press Limited 2011

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References

[1] Argyros, I. K., On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 11 (2004), pp. 103–110.Google Scholar
[2] Argyros, I. K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 298 (2004), pp. 374–397.CrossRefGoogle Scholar
[3] Argyros, I. K., Computational Theory for iterative methods, Studies in Computational Mathematics, Editors: Chui, C.K. and Wyutack, L., Volume 15, Elsevier, New York, USA, 2007.Google Scholar
[4] Dennis, J. E., On the Kantorovich hypotheses for Newton’s method, SIAM J. Numer. Anal., 6 (1969), pp. 493–507.Google Scholar
[5] Deuflhard, P., Newton methods for nonlinear problems, Springer Series in Computational Mathematics, Volume 35, Springer-Verlag, Berlin, 2004.Google Scholar
[6] Gragg, W. B., and Tapia, R. A., Optimal error bounds for the Newton-Kantorovich theorem, SIAM J. Numer. Anal., 11 (1974), pp. 10–13.Google Scholar
[7] Kantorovich, L. V., On Newton’s method for functional equations (Russian), Doklady Akademii Nauk SSSR, 59 (1948), pp. 1237–1240.Google Scholar
[8] Ortega, L. M., and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.Google Scholar
[9] Ostrowski, A. M., Solution of Equations in Euclidean and Banach Spaces, Academic press, New York, 1973.Google Scholar
[10] Potra, F. A., On the a posteriori error estimates for Newton’s method, Beiträge Zur Numer. Math., 12 (1984), pp. 125–138.Google Scholar
[11] Potra, F. A., and Pták, V., Sharp error bounds for Newton’s process, Numer. Math., 34 (1980), pp. 63–72.CrossRefGoogle Scholar
[12] Uko, L. U., and Adeyeye, J. O., Generalized Newton iterative methods for nonlinear operator equations, Nonlinear Studies, 8 (2001), pp. 465–477.Google Scholar
[13] Uko, L. U., and Argyros, I. K., A weak Kantorovich existence theorem for the solution of nonlinear equations, J. Math. Anal. Appl., 2 (2008), pp. 909–914.CrossRefGoogle Scholar
[14] Uko, L. U., and Argyros, I. K., A generalized Kantorovich theorem for nonlinear equations based on function splitting, submitted for publication.Google Scholar
[15] Uko, L. U., and Argyros, I. K., An extension of Argyros’ Kantorovich-type solvability theorem for nonlinear equations, submitted for publication.Google Scholar
[16] Yamamoto, T., Error bounds for Newton’s process derived from the Kantorovich conditions, Japan J. Appl. Math., 2 (1985), pp. 285–292.Google Scholar