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LFANT - 2018

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 θ-functions and the elliptic η-function. They construct short addition sequences reaching an optimal number of N+o(N) 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/(logN)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.