Computing Jacobi’s theta in quasi-linear time

Abstract:

Jacobi’s $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta (z,\tau )$, for $z, \tau$ verifying certain conditions, with precision $P$ in $O(\mathcal {M}(P) \sqrt {P})$ bit operations, where $\mathcal {M}(P)$ denotes the number of operations needed to multiply two complex $P$-bit numbers. We generalize an algorithm which computes specific values of the $\theta$ function (the theta-constants) in asymptotically faster time; this gives us an algorithm to compute $\theta (z, \tau )$ with precision $P$ in $O(\mathcal {M}(P) \log P)$ bit operations, for any $\tau \in \mathcal {F}$ and $z$ reduced using the quasi-periodicity of $\theta$.