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

- 401 Downloads
- 5 Citations

## Abstract

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.

## Preview

Unable to display preview. Download preview PDF.

## References

- Astrand, J. J.: 1890,
*Hülfstafeln*, (Leipzig).Google Scholar - Bauschinger, J.: 1901,
*Tafeln zur Theoretischen Astronomie*, (Leipzig).Google Scholar - Bergam, M. J. and Prussing, J. E.: 1982, 'Comparison of Starting Values for Iterative Solutions to a Universal Kepler's Equation',
*J. Astronaut. Sci.***30**, 75–84.MathSciNetADSGoogle Scholar - Broucke, R.: 1980, 'On Kepler's Equation and Strange Attractors',
*J. Astronaut. Sci.***28**, 255–265.Google Scholar - Conway, B. A.: 1986, 'An Improved Algorithm Due to Laguerre for the Solution of Kepler's Equation',
*Celest. Mech.***39**, 199–211.zbMATHCrossRefADSGoogle Scholar - Conway, B. A.: 1987, 'An Efficient and Unfailing Method for the Solution of Kepler's Equation',
*Proc. 38th Congress of the Int. Astronautical Federation*, Brighton, England.Google Scholar - Danby, J. M. A.: 1987, 'The Solution of Kepler's Equation III',
*Celest. Mech.***40**, 303–312.zbMATHCrossRefADSGoogle Scholar - Danby, J. M. A. and Burkardt, T. M.: 1983, 'The Solution of Kepler's Equation I',
*Celest. Mech.***31**, 95–107.zbMATHCrossRefADSGoogle Scholar - Gooding, R. H. and Odell, A.W.: 1988, 'The Hyperbolic Kepler's Equation (and the Revised Elliptic Equation Revisited)',
*Celest. Mech.***44**, 267–282.zbMATHMathSciNetCrossRefADSGoogle Scholar - Ng, E. W.: 1979, 'A General Algorithm for the Solution of Kepler's Equation for Elliptic Orbits',
*Celest. Mech.***20**, 243–249.zbMATHCrossRefADSGoogle Scholar - Prussing, J. E.: 1977, 'Bounds on the Solution to Kepler's Problem',
*J. Astronaut. Sci.***25**, 123–128.ADSGoogle Scholar - Smart, W. M.: 1931,
*Text-book on Spherical Astronomy*, Cambridge University Press, Cambridge.Google Scholar - Smith, G. R.: 1979, 'A Simple, Efficient Starting Value for the Iterative Solution of Kepler's Equation',
*Celest. Mech.***19**, 163–166.zbMATHCrossRefADSGoogle Scholar - Taff, L. G. and Brennan, T. A.: 1989, 'On Solving Kepler's Equation',
*Celest. Mech.***46**, 163–176.zbMATHMathSciNetCrossRefADSGoogle Scholar