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

@article{Charles1997TheCO,
  title={The Convergence of Newton–Raphson Iteration with Kepler's Equation},
  author={E. D. Charles and Jeremy B. Tatum},
  journal={Celestial Mechanics and Dynamical Astronomy},
  year={1997},
  volume={69},
  pages={357-372}
}
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… 
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References

SHOWING 1-10 OF 12 REFERENCES
A general algorithm for the solution of Kepler's equation for elliptic orbits
An efficient algorithm is presented for the solution of Kepler's equationf(E)=E−M−e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple
On solving Kepler's equation
TLDR
This work attacks Kepler's equation with the unified derivation of all known bounds and several starting values, a proof of the optimality of these bounds, a very thorough numerical exploration of a large variety of starting values and solution techniques, and finally the best and simplest starting value/solution algorithm: M + e and Wegstein's secant modification of the method of successive substitutions.
An improved algorithm due to laguerre for the solution of Kepler's equation
A root-finding method due to Laguerre (1834–1886) is applied to the solution of the Kepler problem. The speed of convergence of this method is compared with that of Newton's method and several
Comparison of starting values for iterative solutions to a universal Kepler's equation
General starting values for the iterative numerical solution of a universal Kepler's equation for position in a conic orbit at a specified time are investigated. Three starting values based on recent
The hyperbolic Kepler equation (and the elliptic equation revisited)
A procedure is developed that, in two iterations, solves the hyperbolic Kepler's equation in a very efficient manner, and to an accuracy that proves to be always better than 10−20 (relative
The solution of Kepler's equation, I
Methods of iteration are discussed in relation to Kepler's equation, and various initial ‘guesses’ are considered, with possible strategies for choosing them. Several of these are compared; the
A simple, efficient starting value for the iterative solution of Kepler's equation
A simple starting value for the iterative solution of Kepler's equation in the elliptic case is presented. This value is then compared against five other starting values for 3750 test cases. In
The solution of Kepler's equation, II
Starting values for the iterative solution of Kepler's equation are considered for hyperbolic orbits, and for generalized versions of the equation, including the use of universal variables.
Text-Book on Spherical Astronomy
1. Spherical trigonometry 2. The celestial sphere 3. Refraction 4. The meridian circle 5. Planetary motions 6. Time 7. Planetary phenomena and heliographic co-ordinates 8. Aberration 9. Parallax 10.
The solution of Kepler's equation, III
Recently proposed methods of iteration and initial guesses are discussed, including the method of Laguerre-Conway. Tactics for a more refined initial guess for use with universal variables over a
...
1
2
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