## Section: New Results

### Computing Jacobi's theta function in quasi-linear time

Participant : Hugo Labrande [contact] .

We designed a new algorithm that improves the complexity of computing the
value of the Jacobi theta function, $\theta (z,\tau )$ to arbitrary
precision [23] . The algorithm uses a
quadratically convergent sequence similar to the complex AGM, as well as
Newton's method; its complexity is $O\left(\mathcal{M}\right(n)logn)$ for
computing the value up to an error bounded by ${2}^{-n}$, which is an improvement
over the state-of-the-art complexity of $O\left(\mathcal{M}\left(n\right)\sqrt{n}\right)$.
Here,
$\mathcal{M}\left(n\right)$ denotes the time taken by a multiplication of two $n$-bit
numbers. We provide bounds on the loss of significant digits incurred during
the computation. The algorithm was implemented using GNU MPC, showing
practical improvement over (our optimized implementation of) existing algorithms for precision above
approximately $300,000$ bits. The paper was submitted to
*Mathematics of Computation*.