Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms

TL;DR

This paper presents algorithms for high-precision numerical computation of elliptic functions, integrals, and modular forms with rigorous error bounds, implemented in the open source Arb library, emphasizing performance for complex variables.

Contribution

It introduces efficient algorithms for arbitrary-precision computation of elliptic functions and related special functions with rigorous error control, and discusses their implementation details.

Findings

Algorithms achieve high precision with rigorous error bounds.

Implementations are available in the open source Arb library.

Performance optimized for tens to thousands of digits of precision.

Abstract

We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for arbitrary complex variables. Implementations in ball arithmetic are available in the open source Arb library. We discuss the algorithms from a concrete implementation point of view, with focus on performance at tens to thousands of digits of precision.