Section: New Results

Computing Jacobi's Theta in Quasi-linear Time

Participant : Hugo Labrande [contact] .

Most of the results have been obtained in 2015. The article was accepted for publication in 2016 [5].

We study the multiprecision computation of the theta function in genus 1, i.e., the Jacobi theta function. The main result is that $\theta (z,\tau )$ can be computed in time that is quasi-linear in the precision $P$, using an algorithm which follows the same strategy as the case of theta-constants (Dupont, 2006). A thorough analysis of the precision loss is given in order to prove correctness.

Along with this work, we have publicly released an open source implementation of the algorithm in C (using the GNU MPC library). This implementation shows this algorithm is faster than a more naive approach for precisions greater than 300,000 digits.