On Computing Jacobi's Elliptic Function \texttt{sn}

TL;DR

This paper introduces a method for computing Jacobi's elliptic function sn efficiently on the period parallelogram, utilizing elliptic integrals and Newton's method, with a focus on accuracy and computational simplicity.

Contribution

It provides a novel approach combining elliptic integrals and Newton's method for accurate computation of sn on the period parallelogram without extensive numerical procedures.

Findings

Efficient computation of sn on the period parallelogram.

Use of Newton's method for fixed m values.

Computation outside the main domain relies solely on function properties.

Abstract

The paper presents a method to compute the Jacobi's elliptic function \texttt{sn} on the period parallelogram. For fixed $m$ it requires first to compute the complete elliptic integrals $K=K(m)$ and $K'=K(1-m).$ The Newton method is used to compute sn(z,m), when $z\in [0,K]\cup[0,i K').$ The computation in any other point does not require the usage of any numerical procedure, it is done only with the help of the properties of sn.