On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds

  • Teles A. FernandesEmail author
  • Orizon P. Ferreira
  • Jinyun Yuan


In this paper, we study Newton’s method for finding a singularity of a differentiable vector field defined on a Riemannian manifold. Under the assumption of invertibility of the covariant derivative of the vector field at its singularity, we show that Newton’s method is well defined in a suitable neighborhood of this singularity. Moreover, we show that the sequence generated by Newton’s method converges to the solution with superlinear rate.


Riemannian manifold Newton’s method Local convergence Superlinear rate 

Mathematics Subject Classification

90C30 49M15 65K05 



The work was supported by FAPEG, UESB, and CNPq Grants 305158/2014-7 and 408151/2016-1.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Teles A. Fernandes
    • 1
    Email author
  • Orizon P. Ferreira
    • 2
  • Jinyun Yuan
    • 3
  1. 1.Universidade Estadual do Sudoeste da BahiaVitória da ConquistaBrazil
  2. 2.IMEUniversidade Federal de GoiásGoiâniaBrazil
  3. 3.Universidade Federal do ParanáCuritibaBrazil

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