THE CONVERGENCE OF NEWTON-RAPHSON ITERATION WITH KEPLERS EQUATION

Conway (Celest. Mech. 39, 199-211, 1986) drew attention to the circums tance that when the Newton-Raphson algorithm is applied to Kepler's eq uation for very high eccentricities there are certain apparently capri cious and random values of the eccentricity and mean anomaly for which convergence seems not to be easily reached when the starting guess fo r the eccentric anomaly is taken to be equal to the mean anomaly. We e xamine this chaotic behavior and show that rapid convergence is always reached if the first guess for the eccentric anomaly is rr. We presen t graphs and an empirical formula for obtaining an even better first g uess. We also examine an unstable situation where iterations oscillate between two incorrect results until the instability results in sudden convergence to the unique correct solution.