Solving Kepler’s equation via Smale’s \(\alpha \)-theory

  • Martín Avendano
  • Verónica Martín-Molina
  • Jorge Ortigas-Galindo
Original Article


We obtain an approximate solution \(\tilde{E}=\tilde{E}(e,M)\) of Kepler’s equation \(E-e\sin (E)=M\) for any \(e\in [0,1)\) and \(M\in [0,\pi ]\). Our solution is guaranteed, via Smale’s \(\alpha \)-theory, to converge to the actual solution \(E\) through Newton’s method at quadratic speed, i.e. the \(n\)-th iteration produces a value \(E_n\) such that \(|E_n-E|\le (\frac{1}{2})^{2^n-1}|\tilde{E}-E|\). The formula provided for \(\tilde{E}\) is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near \(e=1\) and \(M=0\), where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region \([0,1)\times [0,\pi ]\) if only rational functions are allowed in each branch.


Kepler’s equation Smale’s \(\alpha \)-theory Newton’s method Optimal starter 



The authors would like to thank Prof. Antonio Elipe for his valuable help. Second author is partially supported by the MINECO grant MTM2011-22621 and the FQM-327 group (Junta de Andalucía, Spain). Third author is partially supported by the MINECO grant MTM2010-21740-C02-02. Both are also partially supported by the Grupo consolidado E15 “Geometría” (Gobierno de Aragón, Spain) and the “Centro Universitario de la Defensa de Zaragoza” grant ID2013-15.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Martín Avendano
    • 1
    • 2
  • Verónica Martín-Molina
    • 1
    • 2
  • Jorge Ortigas-Galindo
    • 1
    • 2
  1. 1.Centro Universitario de la Defensa SaragossaSpain
  2. 2.IUMAUniversidad de ZaragozaSaragossaSpain

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