Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms
Provides rigorous and efficient numerical methods for a fundamental class of special functions, benefiting researchers in number theory, cryptography, and scientific computing.
The paper presents algorithms for computing elliptic functions, elliptic integrals, and modular forms to arbitrary precision with rigorous error bounds, with implementations in the Arb library. Performance is demonstrated for tens to thousands of digits of precision.
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.