The convergence of Newton-Raphson iteration with Kepler's equation
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.
scientific article; zbMATH DE number 1282932