Section: New Results
Optimal addition sequences for theta functions
Participants : Andreas Enge, Fredrik Johansson.
In , A. Enge, F. Johansson and their coauthor W. Hart consider the problem of numerically evaluating one-dimensional -functions and the elliptic -function. They construct short addition sequences reaching an optimal number of multiplications for evaluating the function as a sparse series with 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 can be written as , where , 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 multiplications for any . These theoretical improvements also lead to an interesting speed-up in practice, and they have been integrated into the CM and the ARB software.