# Procedures for solving Kepler's equation

```@article{Odell1986ProceduresFS,
title={Procedures for solving Kepler's equation},
author={A. W. Odell and Robert H. Gooding},
journal={Celestial mechanics},
year={1986},
volume={38},
pages={307-334}
}```
• Published 1 April 1986
• Mathematics
• Celestial mechanics
We review starting formulae and iteration processes for the solution of Kepler's equation, and give details of two complete procedures. The first has been in use for a number of years, but the second is entirely new. The new procedure operates with an iterative process that always gives fourth-order convergence and is taken to only two iterations. The error in the resulting solution then never exceeds 7×10−15 rad.
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• 2010
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## References

SHOWING 1-10 OF 12 REFERENCES
The solution of Kepler's equation, I
• Physics
• 1983
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
Comparison of starting values for iterative solutions to a universal Kepler's equation
• Physics
• 1982
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
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
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
An exact analytical solution of Kepler's equation
• Physics
• 1972
Complex-variable analysis is used to develop an exact solution to Kepler's equation, for both elliptic and hyperbolic orbits. The method is based on basic properties of canonical solutions to
Universal Keplerian state transition matrix
A completely general method for computing the Keplerian state transition matrix in terms of Goodyear's universal variables is presented. This includes a new scheme for solving Kepler's problem which
On a class of iteration formulas and some historical notes
The class of iteration formulas obtainable by rational approximations of “Euler's formula” is derived with the corresponding error estimates and it is shown how a number of known formulas may be derived from this unified approach.
Neue Formeln und Hilfstafeln zur Ephemeridenrechnung
In einer groseren Abhandlung, die an anderer Stelle erscheinen wird, habe ich eine neue Theorie und Methode der Ephemeridenrechnung entwickelt, deren Anwendung in vielen praktischen Fallen Vorteile