The Convergence of Newton–Raphson Iteration with Kepler's Equation

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.