CORDIC-like method for solving Kepler’s equation

Institut für Astrophysik, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
e-mail: zechmeister@astro.physik.uni-goettingen.de

Received: 4 April 2018
Accepted: 14 August 2018

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 also applicable for the hyperbolic case.

Methods. Based on the idea of CORDIC, our method 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. Half or two-thirds of the iterations can be saved in combination with Newton’s or Halley’s method at the cost of one division.