A simplified proof of the Kantorovich theorem for solving equations using telescopic series

Authors

  • Ioannis K. Argyros Cameron University, USA
  • Hongmin Ren Hangzhou Polytechnic, China

DOI:

https://doi.org/10.33993/jnaat442-1084

Keywords:

Newton-Kantorovich method, Banach space, majorizing series, telescopic series, Kantorovich theorem
Abstract views: 305

Abstract

We extend the applicability of the Kantorovich theorem (KT) for solving nonlinear equations using Newton-Kantorovich method in a Banach space setting. Under the same information but using elementary scalar telescopic majorizing series, we provide a simpler proof for the (KT) [2], [7]. Our results provide at least as precise information on the location of the solution. Numerical examples are also provided in this study.

Downloads

Download data is not yet available.

Author Biography

Ioannis K. Argyros, Cameron University, USA

Full tenured Professor of Mathematics.

References

I.K. Argyros, Polynomial equations in abstract spaces and applications, St. Lucie/CRC/Lewis Publ. Mathematics Series, 1998, Boca Raton Florida.

I.K. Argyros, Computational theory of iterative methods, Series: Studies in Computational Mathematics 15, Editors, C.K. Chui and L. Wuytack, Elservier Publ. Co. New York, USA, 2007.

S. Chandrasekhar, Radiative Transfer, Dover. Publ, 1960, New York.

P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer-Verlag, Berlin, Heidelberg, 2004.

J.A. Ezquerro, M.A. Hern´andez, New Kantorovich-type conditions for Halley’s method, Appl. Num. Anal. Comp. Math., 2 (2005), pp. 70–77, http://doi.org/10.1002/anac.200410024 DOI: https://doi.org/10.1002/anac.200410024

J.A. Ezquerro, M.A. Hernandez, An optimization of Chebyshev-method, J. Complexity, 25 (2009), pp. 343–361, http://doi.org/10.1016/j.jco.2009.04.001 DOI: https://doi.org/10.1016/j.jco.2009.04.001

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

J.H. Mathews, Bibligraphy for Newton’s method, Available from: http://mathfaculty.fullerton.edu/mathews/n2003/newtonsmethod/Newton’sMethodBib/Links/Newton’sMethodBiblnk3.html.

Yu. Nesterov, Introductory Lectures on Convex Programming, Kluwer, Boston, 2004. DOI: https://doi.org/10.1007/978-1-4419-8853-9

S.J. Wright, Primal-Dual Interior Point Methods, SIAM, Philadelphia, 1997. DOI: https://doi.org/10.1137/1.9781611971453

T.J. Ypma, Historical developments of the Newton-Raphson method, SIAM Review, 37 (1995), pp. 531–551, https://doi.org/10.1137/1037125 DOI: https://doi.org/10.1137/1037125

Downloads

Published

2015-12-31

How to Cite

Argyros, I. K., & Ren, H. (2015). A simplified proof of the Kantorovich theorem for solving equations using telescopic series. J. Numer. Anal. Approx. Theory, 44(2), 146–153. https://doi.org/10.33993/jnaat442-1084

Issue

Vol. 44 No. 2 (2015)

Section

Articles

License

Copyright (c) 2016 Journal of Numerical Analysis and Approximation Theory

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funding data