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Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds

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  • Bittencourt, Tiberio
  • Ferreira, Orizon Pereira

Abstract

A local convergence analysis of Inexact Newton’s method with relative residual error tolerance for finding a singularity of a differentiable vector field defined on a complete Riemannian manifold, based on majorant principle, is presented in this paper. We prove that under local assumptions, the Inexact Newton method with a fixed relative residual error tolerance converges Q linearly to a singularity of the vector field under consideration. Using this result we show that the Inexact Newton method to find a zero of an analytic vector field can be implemented with a fixed relative residual error tolerance. In the absence of errors, our analysis retrieves the classical local theorem on the Newton method in Riemannian context.

Suggested Citation

  • Bittencourt, Tiberio & Ferreira, Orizon Pereira, 2015. "Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian manifolds," Applied Mathematics and Computation, Elsevier, vol. 261(C), pages 28-38.
    • Handle: RePEc:eee:apmaco:v:261:y:2015:i:c:p:28-38
    DOI: 10.1016/j.amc.2015.03.080
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    References listed on IDEAS

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    1. Jin-Hua Wang & Jen-Chih Yao & Chong Li, 2012. "Gauss–Newton method for convex composite optimizations on Riemannian manifolds," Journal of Global Optimization, Springer, vol. 53(1), pages 5-28, May.
    2. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
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    Cited by:

    1. Davide Cuccato & Alessandro Saccon & Antonello Ortolan & Alessandro Beghi, 2016. "Computing Laser Beam Paths in Optical Cavities: An Approach Based on Geometric Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 297-315, October.
    2. Petre Birtea & Dan Comănescu, 2017. "Newton Algorithm on Constraint Manifolds and the 5-Electron Thomson Problem," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 563-583, May.
    3. Fabiana R. Oliveira & Fabrícia R. Oliveira, 2021. "A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 259-273, July.

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