• The Inria's Research Teams produce an annual Activity Report presenting their activities and their results of the year. These reports include the team members, the scientific program, the software developed by the team and the new results of the year. The report also describes the grants, contracts and the activities of dissemination and teaching. Finally, the report gives the list of publications of the year.

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## 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 $O\left(N/{\left(logN\right)}^{r}\right)$ 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.