Comment on "An efficient code to solve the Kepler equation. Elliptic case"

TL;DR

This paper critically evaluates a recent method for solving the elliptic Kepler equation, correcting misconceptions about achievable accuracy and providing limits for the error based on eccentricity, with implications for designing more precise algorithms.

Contribution

It analytically and numerically demonstrates the true accuracy limits of Newton-Raphson methods for Kepler's equation, correcting prior claims and extending the analysis to hyperbolic cases.

Findings

The accuracy limit is approximately ε/√(2(1-e)) for high eccentricities.

Errors diverge as eccentricity approaches 1, contrary to previous claims.

Provides guidelines for developing more accurate Kepler equation solvers.

Abstract

In a recent MNRAS article, Raposo-Pulido and Pelaez (RPP) designed a scheme for obtaining very close seeds for solving the elliptic Kepler Equation with the classical and the modified Newton-Rapshon methods. This implied an important reduction in the number of iterations needed to reach a given accuracy. However, RPP also made strong claims about the errors of their method that are incorrect. In particular, they claim that their accuracy can always reach the level $\sim5\varepsilon$, where $\varepsilon$ is the machine epsilon (e.g. $\varepsilon=2.2\times10^{-16} $ in double precision), and that this result is attained for all values of the eccentricity $e<1$ and the mean anomaly $M\in[0,\pi]$, including for $e$ and $M$ that are arbitrarily close to $1$ and $0$, respectively. However, we demonstrate both numerically and analytically that any implementation of the classical or modified Newton-Raphson methods for Kepler's equation, including those described by RPP, have a limiting accuracy of the order $\sim\varepsilon/\sqrt{2(1-e)}$. Therefore the errors of these implementations diverge in the limit $e\to1$, and differ dramatically from the incorrect results given by RPP. Despite these shortcomings, the RPP method can provide a very efficient option for reaching such limiting accuracy. We also provide a limit that is valid for the accuracy of any algorithm for solving Kepler equation, including schemes like bisection that do not use derivatives. Moreover, similar results are also demonstrated for the hyperbolic Kepler Equation. The methods described in this work can provide guidelines for designing more accurate solutions of the elliptic and hyperbolic Kepler equations.