Chebyshev–Halley's method on Riemannian manifolds

R. A. Castro*, J. C. Rodríguez, W. W. Sierra, G. L. Di Giorgi, S. J. Gómez

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

This paper is concerned with the problem of finding singular points of vector fields on Riemannian manifolds. We, using non trivial techniques and results of differential geometry, will exhibit a new method defined on complete Riemannian manifolds, which generalizes the important Newton and Chebyshev–Halley's methods. Moreover, a characterization of the convergence under Kantorovich-type conditions and error estimates are also given in this study. Using the method introduced in this paper we give an algorithm which will allow us to find singular points of a vector field on the two-dimensional sphere S2. Finally, in order to illustrate our method and its relevance, we develop an example which allows us to find a singularity of a vector field on S2, such a singularity is not possible to find it using the classical numerical methods on R3.

Original languageEnglish
Pages (from-to)30-53
Number of pages24
JournalJournal of Computational and Applied Mathematics
Volume336
DOIs
StatePublished - Jul 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017 Elsevier B.V.

Keywords

  • Kantorovich-type conditions
  • Newton and Chebyshev–Halley's methods
  • Riemannian manifolds

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