Scaling of the various integration methods as a function of the number of particles (N). The classic Newton integrations scale as ∝N², as indicated with the solid and dashed dark blue lines. The pairwise first expansion scales similarly, but tends to be slightly slower than the pure Newton expansion. The Einstein-Infeld Hoffmann equations to first order scale ∝N³, making large calculations that include the cross-terms unpractical. The scaling presented for Brutus is based on the calculations presented here. The bullet points indicate the mean timescale for acquiring a converged solution, and the line pointed upward ends at the single most expensive calculation in our sample of simulations for that particular N. For ph4, we included the regular implementation as well as the GPU-enabled version (to the right), running on an Intel Xeon CPU E5620 operating at 2.40 GHz and NVIDIA G96 (Quadro FX580), running on a generic 64-bit Ubuntu Linux kernel 2.6.35-32.