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}
}
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. 
Solving Kepler's equation with high efficiency and accuracy
We present a method for solving Kepler's equation for elliptical orbits that represents a gain in efficiency and accuracy compared with those currently in use. The gain is obtained through a starter
Appropriate Starter for Solving the Kepler's Equation
TLDR
This article focuses on the methods that have been used for solving the Kepler’s equation for thirty years, then Kepler's equation will be solved by Newton-Raphson's method, and one appropriate choice first guesses that increase the isotropy and decrease the time of Implementation of solving is introduced.
On solving Kepler's equation for nearly parabolic orbits
We deal here with the efficient starting points for Kepler's equation in the special case of nearly parabolic orbits. Our approach provides with very simple formulas that allow calculating these
Solution of Kepler’s Equation with Machine Precision
Abstract— An algorithm for the numerical solution of Kepler’s equation with machine precision is presented. The convergence of the iterative sequence of Newton’s method is proved for the indicated
A cubic approximation for Kepler's equation
We derive a new method to obtain an approximate solution for Kepler's equation. By means of an auxiliary variable it is possible to obtain a starting approximation correct to about three figures. A
Kepler Equation solver
TLDR
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation, and requires only four transcendental function evaluations.
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
Sequential solution to Kepler’s equation
TLDR
Seven sequential starter values for solving Kepler’s equation are proposed for fast orbit propagation and obtain improved accuracy at lower computational cost as compared to the best existing methods.
Numerical solution of the two-body problem for orbital motion is heavily dependent on efficient solution of Kepler's Equation
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four
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
3
4
...

References

SHOWING 1-10 OF 12 REFERENCES
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
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
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
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
TLDR
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
...
1
2
...