Celestial Mechanics and Dynamical Astronomy

, Volume 129, Issue 4, pp 415–432 | Cite as

An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

  • A. Elipe
  • J. I. Montijano
  • L. Rández
  • M. Calvo
Original Article


In this note a study of the convergence properties of some starters \( E_0 = E_0(e,M)\) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter \( E_0(e,M)\) a set of values \( (e,M) \in [0, 1) \times [0, \pi ]\) that lead to the q-convergence in the sense that Newton’s sequence \( (E_n)_{n \ge 0}\) generated from \( E_0 = E_0(e,M)\) is well defined, converges to the exact solution \(E^* = E^*(e,M)\) of KE and further \( \vert E_n - E^* \vert \le q^{2^n -1}\; \vert E_0 - E^* \vert \) holds for all \( n \ge 0\). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s \(\alpha \)-test with \(q=1/2\). Also since in KE the convergence rate of Newton’s method tends to zero as \( e \rightarrow 0\), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with \( q = e^k \; \widetilde{q} \) for all \( e \in [0,1)\) and a fixed \( \widetilde{q} \le 1\). Some remarks on the use of this theorem to derive a priori estimates of the error \( \vert E_n - E^* \vert \) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.


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



This work has been supported by the Spanish Ministry of Economy, Projects ESP2013-44217-R, DGI MTM2013-47318-C2-1-P.


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Dpto. Matemática Aplicada - IUMAUniversidad de ZaragozaSaragossaSpain
  2. 2.Centro Universitario de la DefensaSaragossaSpain

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