Convergence of starters for solving Kepler’s equation via Smale’s \(\alpha \)-test
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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.
KeywordsKepler’s equation Optimal starters Smale’s \(\alpha \)-test Convergence
- 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
- Dedieu, J.P.: Points fixes, zeros et la mèthode de Newton. In: Mathematics and Applications, vol. 54. Springer, Berlin (2006)Google Scholar
- Deuflhard, P.: Newton methods for nonlinear problems. In: Springer Series in Computational Mathematics, vol. 35. Springer, New York (2004)Google Scholar
- 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