Celestial Mechanics and Dynamical Astronomy

, Volume 127, Issue 1, pp 19–34 | Cite as

Convergence of starters for solving Kepler’s equation via Smale’s \(\alpha \)-test

Original Article
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Abstract

In this note, by using Smale’s \(\alpha \)-theorem on the convergence of Newton’s method, the \(\alpha \)-sets of convergence of some starters of solving the elliptic Kepler’s equation are derived. For each starter we compute the exact \(\alpha \)-set in the eccentricity-main anomaly \((e,M)\in [0,1)\times [0,\pi ]\), showing that these sets are larger than those derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014). Further, new convergence tests based on the Newton–Kantorowitch theorem are given comparing with the derived from Smale’s \(\alpha \)-test.

Keywords

Kepler’s equation Optimal starters Smale’s \(\alpha \)-test Convergence 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Dpto. Matemática Aplicada - IUMAUniversidad de ZaragozaZaragozaSpain
  2. 2.Centro Universitario de la DefensaZaragozaSpain

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