commonly accepted to be sufficient in astrophysical contexts [@CM], requires a tolerance parameter $\theta=\,$const$\,\simeq0.65$ or $\theta_{\rm
        min}\simeq0.5$.

The Total Force Error {#sec:error}
---------------------

The important question here is for which choices of $\theta$ is the approximation error negligible compared to the estimation error? To answer this question, we have performed some experiments using samples of $N=10^4$, $10^5$ and $10^6$ bodies drawn from a Hernquist model and computed the mean averaged squared error (MASE) of the force (per unit mass) [@AT]: $$\label{mase}
  {\rm MASE}({{\mbox{\boldmath$F$}}}) = \left\{{1\over{\sf M_{\rm tot}}} \sum_i \mu_i
    \big(\hat{{{\mbox{\boldmath$F$}}}}_i - {{\mbox{\boldmath$F$}}}({{\mbox{\boldmath$x$}}}_i)\big)^2 \right\}.$$ Here, $\hat{{{\mbox{\boldmath$F$}}}}_i$ and ${{\mbox{\boldmath$F$}}}({{\mbox{\boldmath$x$}}})$ are, respectively, the approximately estimated force for the $i$th body and the true force field of the stellar system. The curly brackets denote the ensemble average over many possible random realizations of the same underlying stellar density by $N$ bodies. We used $10^7/N$ ensembles and computed the MASE$({{\mbox{\boldmath$F$}}})$[^3] for various values of $\theta$, but always at optimum $\epsilon$ [@Db]. As one might already have guessed from the behavior of the approximation error in [Figure \[fig:error\]]{}, the relative increase of the MASE$({{\mbox{\boldmath$F$}}})$ is negligible: even for $\theta=0.9$, the approximation error contributes less than one per cent, in agreement with the findings of [@AT]. Based 