Section: New Results

Optimal addition sequences for theta functions

Participants : Andreas Enge, Fredrik Johansson.

In [20], A. Enge, F. Johansson and their coauthor W. Hart consider the problem of numerically evaluating one-dimensional $\theta$-functions and the elliptic $\eta$-function. They construct short addition sequences reaching an optimal number of $N+o\left(N\right)$ multiplications for evaluating the function as a sparse series with $N$ terms. The proof relies on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, they show that every generalised pentagonal number $c>5$ can be written as $c=2a+b$, where $a$, $b$ are smaller generalised pentagonal numbers. They then give a baby-step giant-step algorithm that breaks through the theoretical barrier achievable with addition sequences, and which uses only $O(N/{\left(logN\right)}^r)$ multiplications for any $r>0$. These theoretical improvements also lead to an interesting speed-up in practice, and they have been integrated into the CM and the ARB software.