The convergence of Newton-Raphson iteration with Kepler's equation
type: Item
instance of: scholarly article
title: The convergence of Newton-Raphson iteration with Kepler's equation
zbMATH Open document ID: 0947.70003
DOI: 10.1023/A:1008200607490
author: E. D. Charles
author: J. B. Tatum
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 2000-11-02T00:00:00Z
review text: The authors examine the chaotic behavior of Newton-Raphson method applied to Kepler's equation for very high eccentricities with some starting values. It is shown that the convergence of Newton-Raphson method can be always obtained for an initial approximation \(E_0\) of eccentric anomaly \(E\) chosen within the interval between the desired root of the equation and \(\pi\). Moreover, the choice of \(E_0=\pi\) it yields successful convergence for any combination of mean anomaly \(M\) and eccentricity \(e\). The authors also show that a modification of Newton-Raphson method results in convergence from a usually taken starting value of \(E_0=M\) even in those cases where chaotic behavior is encountered in the conventional Newton-Raphson iterations. An interesting empirical formula is given for a first approximation \(E_0\) which always yields extremely rapid convergence. Finally, the authors examine an unstable situation where iterations oscillate between two incorrect results, until the instability results in a sudden convergence to the unique correct solution. In the reviewer's opinion, the paper is very interesting from both theoretical and practical point of view.
Mathematics Subject Classification ID: 70-08
Mathematics Subject Classification ID: 70F05
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 1282932
zbMATH Keywords: Newton-Raphson method
zbMATH Keywords: Kepler's equation
zbMATH Keywords: convergence
zbMATH Keywords: initial approximation
zbMATH Keywords: chaotic behavior
reviewed by: Andrzej Marciniak
MaRDI profile type: Publication
full work available at URL: a:1008200607490
OpenAlex ID: W1651774721
Wikidata QID: Q61893623
description: scientific article; zbMATH DE number 1282932