A couple years ago I wrote a blog post on Kepler’s equation
M + e sin E = E.
Given mean anomaly M and eccentricity e, you want to solve for eccentric anomaly E.
There is a simple way to solve this equation. Define
f(E) = M + e sin E
and take an initial guess at the solution and stick it into f. Then take the output and stick it back into f, over and over, until you find a fixed point, i.e. f(E) = E.
The algorithm above is elegant, and practical if you only need to do it once. However, if you need to solve Kepler’s equation billions of times, say in the process of tracking satellite debris, this isn’t fast enough.
An obvious improvement would be to use Newton’s root-finding method rather than the simple iteration scheme above, and this isn’t far from the state of the art. However, there have been improvements over Newton’s method, and a paper posted on arXiv this week gives an algorithm that is about 3 times faster than Newton’s method .
Patterns in applied math
This paper is an example of a common pattern in applied math. It starts with a simple problem that has a simple solution, but this simple solution doesn’t scale. And so we apply advanced mathematics to a problem formulated in terms of elementary mathematics.
In particular, the paper makes use of contour integration. This seems like a step backward in two ways.
First, we have a root-finding problem, but you want to turn it into an integration problem?! Isn’t root-finding faster than integration? Not in this case.
Second, not only are we introducing integration, we’re introducing integration in the complex plane. Isn’t complex analysis complex? Not in the colloquial sense. The use of “complex” as a technical term is unfortunate because complex analysis often simplifies problems. As Jacques Hadamard put it,
The shortest path between two truths in the real domain passes through the complex domain.
 Oliver H. E. Philcox, Jeremy Goodman, Zachary Slepian. Kepler’s Goat Herd: An Exact Solution for Elliptical Orbit Evolution. arXiv:2103.15829