Solving Kepler's equation CORDIC-like

TL;DR

This paper introduces a CORDIC-like algorithm for efficiently computing the eccentric anomaly and its sine and cosine without transcendental functions, applicable to both elliptical and hyperbolic Kepler's equations.

Contribution

It presents a novel, simple, and efficient method based on CORDIC that avoids transcendental functions for solving Kepler's equations, with adjustable precision.

Findings

Linear convergence for all mean anomalies and eccentricities

Single precision achieved with 29 iterations, double with 55

Potential to reduce iterations using Newton's or Halley's method

Abstract

Context. Many algorithms to solve Kepler's equations require the evaluation of trigonometric or root functions. Aims. We present an algorithm to compute the eccentric anomaly and even its cosine and sine terms without usage of other transcendental functions at run-time. With slight modifications it is applicable for the hyperbolic case, too. Methods. Based on the idea of CORDIC, it requires only additions and multiplications and a short table. The table is independent of eccentricity and can be hardcoded. Its length depends on the desired precision. Results. The code is short. The convergence is linear for all mean anomalies and eccentricities e (including e = 1). As a stand-alone algorithm, single and double precision is obtained with 29 and 55 iterations, respectively. One half or two third of the iterations can be saved in combination with Newton's or Halley's method at the cost of one division.