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

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


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.


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



This work has been supported by the Spanish Ministry of Economy, Projects BFM2001-2562, ESP2013-44217-R, DGI MTM2013-47318-C2-1-P and by the Aragon Government and European Social Fund (Groups E-48 and E-65).


  1. Argyros, I.K., Hilout, S., Khattri, S.K.: Expanding the applicability of Newton’s method using Smale’s \(\alpha \)-theory. J. Comput. Appl. Math. 261, 183–200 (2014)Google Scholar
  2. Avendaño, M., Martín-Molina, V., Ortigas-Galindo, J.: Solving Kepler’s equation via Smale’s \(\alpha \)-theory. Celest. Mech. Dyn. Astron. 119, 27–44 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. Avendaño, M., Martín-Molina, V., Ortigas-Galindo, J.: Approximate solutions of the hyperbolic Kepler equation. Celest. Mech. Dyn. Astron. 123, 435–451 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. AIAA Education Series, Reston (1999)CrossRefzbMATHGoogle Scholar
  5. Calvo, M., Elipe, A., Montijano, J.I., Rández, L.: Optimal starters for solving the elliptic Kepler’s equation. Celest. Mech. Dyn. Astron. 115, 143–160 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. Colwell, P.: Solving Kepler’s Equation Over Three Centuries. Willmann-Bell, Richmond (1993)zbMATHGoogle Scholar
  7. Davis, J.J., Mortari, D., Bruccoleri, C.: Sequential solution to Kepler’s equation. Celest. Mech. Dyn. Astron. 108, 59–72 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. Dedieu, J.P.: Points fixes, zeros et la mèthode de Newton. In: Mathematics and Applications, vol. 54. Springer, Berlin (2006)Google Scholar
  9. Deuflhard, P.: Newton methods for nonlinear problems. In: Springer Series in Computational Mathematics, vol. 35. Springer, New York (2004)Google Scholar
  10. Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)zbMATHGoogle Scholar
  11. Mortari, D., Elipe, A.: Solving Kepler’s equation using implicit functions. Celest. Mech. Dyn. Astron. 118, 1–11 (2014)ADSMathSciNetCrossRefGoogle Scholar
  12. Rheinbold, W.C.: On a Theorem of S. Smale’about Newton’s method for analytic mappings. Appl. Math. Lett. 1, 69–72 (1988)MathSciNetCrossRefGoogle Scholar
  13. Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R.E. et al. (eds.) The Merging of Disciplines in Pure, Applied and Computational Mathematics. Springer, New York, pp. 185–196 (1986)Google Scholar
  14. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1983)zbMATHGoogle Scholar
  15. Vallado, D.A.: Fundamentals of Astrodynamics and Applications, vol. 2. McGraw-Hill, New York (2001)zbMATHGoogle Scholar
  16. Wang, D., Zhao, F.: The theory of Smale’s point estimation and its applications. J. Comput. Appl. Math. 60, 253–269 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

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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