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.
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Acknowledgments
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).
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Calvo, M., Elipe, A., Montijano, J.I. et al. Convergence of starters for solving Kepler’s equation via Smale’s \(\alpha \)-test. Celest Mech Dyn Astr 127, 19–34 (2017). https://doi.org/10.1007/s10569-016-9713-0
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DOI: https://doi.org/10.1007/s10569-016-9713-0