on this result, one may advocate the usage of tolerance parameters larger than $\theta_{\rm
  min}\simeq0.5$. However, the distribution of approximation errors is not normal, and the rms error, which is essentially measured by (the square root of) the MASE$({{\mbox{\boldmath$F$}}})$, may well underestimate the danger of using large $\theta$. We therefore cannot recommend using $\theta_{\rm min}{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}0.7$.

Performance Tests {#sec:perform}
=================

Scaling with $N$ {#sec:scaling}
----------------

We measured the CPU time consumption for both constant $\theta=0.65$ and $\theta=\theta({{\sf M}^{}})$ with $\theta_{\rm min}=0.5$, using $s=6$ in either case. [Figure \[fig:N\]]{} plots the consumed time (averaged over many experiments) per body versus $N$ plotted on a logarithmic scale. For the case of $\theta=\theta({{\sf M}^{}})$, [Table \[tab:N\]]{} gives the number of cells as well as the number of individual body-body (B-B), cell-body (C-B) and cell-cell (C-C) interactions, where the latter two are split into those done via a Taylor expansion and direct summation (subscripts ‘app’ and ‘dir’), respectively.

[@rrrrrrrr]{} $N$ & $N_{\rm cells}$ & B-B & C-B$_{\rm app}$ & C-B$_{\rm dir}$ & C-C$_{\rm app}$ & C-C$_{\rm dir}$ & total 1000 & 370 & 210 & 3746 & 2049 & 4057 & 1600 & 11662 3000 & 1090 & 472 & 16870 & 5184 & 30101 