Greengard's N-Body Algorithm is not Order N

Greengard's $N$-body algorithm claims to compute the pairwise interactions in a system of $N$ particles in $O(N)$ time for a fixed precision. In this paper, we show that the choice of precision is not independent of $N$ and has a lower bound of $\log N$. We use this result to show that Greengard's algorithm is not $O(N)$.