The Convergence of Newton–Raphson Iteration with Kepler's Equation |
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| Authors: | E D Charles J B Tatum |
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| Institution: | (1) 16 Greenfield Crescent, Brighton, U.K;(2) Department of Physics and Astronomy, University of Victoria, Canada |
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| Abstract: | Conway (Celest. Mech. 39, 199–211, 1986) drew attention to the circumstance that when the Newton–Raphson algorithm is applied
to Kepler's equation for very high eccentricities there are certain apparently capricious and random values of the eccentricity
and mean anomaly for which convergence seems not to be easily reached when the starting guess for the eccentric anomaly is
taken to be equal to the mean anomaly. We examine this chaotic behavior and show that rapid convergence is always reached
if the first guess for the eccentric anomaly is π. We present graphs and an empirical formula for obtaining an even better
first guess. We also examine an unstable situation where iterations oscillate between two in correct results until the instability
results in sudden convergence to the unique correct solution.
This revised version was published online in July 2006 with corrections to the Cover Date. |
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| Keywords: | Kepler's equation |
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