Almost closed interscribed polygons

TL;DR

This paper introduces a novel iterative algorithm for computing ratios of elliptic integrals using properties of almost closed polygons interscribed between conics, offering high precision and stability.

Contribution

It presents a new approach leveraging polygon properties to compute elliptic integral ratios, distinct from traditional methods, with potential broader applications.

Findings

Algorithm achieves high precision in elliptic integral ratio computation.

Iterative process is numerically stable and comparable in speed to the arithmetic-geometric mean method.

Some iteration numbers correspond to denominators of convergent fractions for elliptic integral ratios.

Abstract

We demonstrate a new approach to the computation of ratios of elliptic integrals. It turns out that almost closed polygons interscribed between two conics retain some of the properties of such closed polygons. We apply these retained properties to compute ratios of an incomplete elliptic integral over the complete one. This computation is based on an iterative procedure to determine the sequence of vertices of a polygon interscribed between two conics. Surprisingly, some iteration numbers are the denominators of the convergent fractions for thye ratio of some elliptic integrals. The algorithm ensures high precision, is numerically stable and as fast as the arithmetic-geometric mean method, though not faster. Nonetheless, there are reasons to consider the proposed algorithm as it is quite differentfrom other integration methods and may be applicable to other problems.