A fast multipole method for stellar dynamics

Table 2 Timings and errors for FMM runs for different N

N	$t_{tree}$	$t_{ffmm}$	$t_{up}$	$t_{walk}$	$t_{down}$	${δ a / f}_{rms}$	${δ a / f}_{99.99 %}$
10⁴	0.0004	0.0011	0.0012	0.0317	0.0005	2.81 × 10⁻⁷	1.32 × 10⁻⁶
10⁵	0.0019	0.0042	0.0097	0.2131	0.0044	3.61 × 10⁻⁷	2.51 × 10⁻⁶
10⁶	0.0290	0.0245	0.0957	1.5366	0.0431	3.85 × 10⁻⁷	3.32 × 10⁻⁶
10⁷	0.2559	0.1897	0.8870	11.358	0.3999	4.05 × 10⁻⁷	2.20 × 10⁻⁶

The runs used p = 10 and ϵ = 10^−6.25. The timings are given in seconds and refer to, respectively, the tree building; the estimation of f via low-order FMM; the passing up of z, s, min{f}, and $M_{n}^{m}$ ; the dual tree walk; and the passing down of $F_{n}^{m}$ and evaluation of gravity. See also Figure 11.