Newton's method in Riemannian manifolds

Authors

  • Ioannis K. Argyros Cameron University, USA

DOI:

https://doi.org/10.33993/jnaat372-883

Keywords:

Newton's method, Riemannian manifold, local/semilocal convergence, singularity of a vector field, Newton-Kantorovich method
Abstract views: 254

Abstract

Using more precise majorizing sequences than before [1], [8], and under the same computational cost, we provide a finer semilocal convergence analysis of Newton's method in Riemannian manifolds with the following advantages: larger convergence domain, finer error bounds on the distances involved, and a more precise information on the location of the singularity of the vector field.

Downloads

Download data is not yet available.

References

Alvarez, F., Bolte, J. and Munier, J., A unifying local convergence result for Newton's method in Riemannian manifolds, Institut National de Recherche en informatique et en automatique, Theme Num-Numeriques, Project, Sydoco, Rapport de recherche No. 5381, November 2004, France.

Argyros, I. K., An improved convergence analysis and applications for Newton-like methods in Banach space, Numer. Funct. Anal. Optim., 24, nos. 7-8, pp. 653-672, 2003, https://doi.org/10.1081/nfa-120026364 DOI: https://doi.org/10.1081/NFA-120026364

Argyros, I. K., A unifying local-semilocal convergence and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Applic., 298, pp. 374-397, 2004, https://doi.org/10.1016/j.jmaa.2004.04.008 DOI: https://doi.org/10.1016/j.jmaa.2004.04.008

Argyros, I. K., On the Newton-Kantorovich method in Riemannian manifolds, Advances in Nonlinear Variational Inequalities, 8, no. 2, pp. 81-85, 2005.

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

Argyros, I. K., On a class of Newton-like methods for solving nonlinear equations, J. Comput. Appl. Math., https://doi.org/10.1016/j.cam.2008.08.042 DOI: https://doi.org/10.1016/j.cam.2008.08.042

Do Carano, M., Riemannian Geometry, Birkhäuser, Boston, 1992, https://doi.org/10.1007/978-1-4757-2201-7 DOI: https://doi.org/10.1007/978-1-4757-2201-7

Ferreira, O. P. and Svaiter, B. F., Kantorovich's theorem on Newton's method in Riemannian manifolds, J. Complexity, 18, pp. 304-353, 2002, https://doi.org/10.1006/jcom.2001.0582 DOI: https://doi.org/10.1006/jcom.2001.0582

Kantorovich, L. V. and Akilov, G. P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982. DOI: https://doi.org/10.1016/B978-0-08-023036-8.50010-2

Zabrejko, P. P. and Nguen, D. F., The majorant method in the theory of Newton-Kantorovich approximations and the Ptak error estimates, Numer. Funct. Anal. Optim., 9, pp. 671-674, 1987, https://doi.org/10.1080/01630568708816254 DOI: https://doi.org/10.1080/01630568708816254

Downloads

Published

2008-08-01

How to Cite

Argyros, I. K. (2008). Newton’s method in Riemannian manifolds. Rev. Anal. Numér. Théor. Approx., 37(2), 119–125. https://doi.org/10.33993/jnaat372-883

Issue

Section

Articles