Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms

Fredrik Johansson (1)

1 LFANT - Lithe and fast algorithmic number theory

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.

Domaines: Logiciel mathématique [cs.MS]; Analyse numérique [cs.NA]

Fredrik Johansson. Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms. Johannes Blümlein; Carsten Schneider; Peter Paule. Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Springer, pp.269-293, 2019, Texts & Monographs in Symbolic Computation. ⟨hal-01817952⟩