Journal of Applied Mathematics and Computing

, Volume 25, Issue 1–2, pp 345–351 | Cite as

An improved unifying convergence analysis of Newton’s method in Riemannian manifolds

  • Ioannis K. Argyros


Using more precise majorizing sequences we provide a finer convergence analysis than before [1], [7] of Newton’s method in Riemannian manifolds with the following advantages: weaker hypotheses, finer error bounds on the distances involved and a more precise information on the location of the singularity of the vector field.

AMS Mathematics Subject Classification

65H10 65B05 65G99 47H17 49M15 

Key words and phrases

Newton’s method Riemannian manifold local/semilocal convergence singularity of a vector field Newton-Kantorovich method 


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  1. 1.
    F. Alvarez, J. Bolte, and J. Munier,A unifying local convergence result for Newton’s method in Riemannian manifolds, Institut National de Recherche en informatique et en avtomatique, Theme Num-Numeriques, Project, Sydoco, Rapport de recherche No. 5381, November 2004, Cedex, France.Google Scholar
  2. 2.
    I.K. Argyros,An improved convergence analysis and applications for Newton-like methods in Banach space, Numer. Funct. Anal. Optim.24, 7 and 8 (2003), 653–672.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    I.K. Argyros,A unifying local-semilocal convergence and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Applic.298 (2004), 374–397.MATHCrossRefGoogle Scholar
  4. 4.
    I.K. Argyros,On the Newton-Kantorovich method in Riemannian manifolds, Advances in Nonlinear Variational Inequalities,8, 2 (2005), 81–85.MATHMathSciNetGoogle Scholar
  5. 5.
    I.K. Argyros,Newton Methods, Nova Science Publ. Corp., New York, 2005.MATHGoogle Scholar
  6. 6.
    M. Do Carano,Riemannian Geometry, Birkhäuser, Boston, 1992.Google Scholar
  7. 7.
    O.P. Ferreira and B.F. Svaiter,Kantorovich’s theorem on Newton’s method in Riemannian manifolds, J. Complexity,18 (2002), 304–353.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    L.V. Kantorovich and G.P. Akilov,Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982.Google Scholar
  9. 9.
    P.P. Zabrejko and D.F. Nguen,The majorant method in the theory of Newton-Kantorovich approximations and the Ptak error estimates, Numer. Funct. Anal. Optim.9 (1987), 671–674.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational & Applied Mathematics and Korean SIGCAM 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCameron UniversityLawtonUSA

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