Figure 9

Error-cost relation. Wall-clock time versus relative (top) and scaled (bottom) acceleration error for $N={10}^7$ particles drawn from a Plummer sphere. The top panel reports runs using multipole-acceptance criterion (16a) with a low-order-FMM estimate for $a_b$, while the bottom panel reports runs using multipole-acceptance criterion (16b) with a low-order-FMM estimate for $f_b$. Each pair of open and filled symbols (of same colour and ordinate) refers to another FMM run with expansion order p as indicated and a different value for parameter ϵ in equation (16). The timings include all phases of the computation, including tree building and low-order estimation of $a_b$ or $f_b$ - for comparison, the direct-summation calculation for obtaining the ‘true’ accelerations took 25k seconds on the same hardware. The thin dotted and dashed lines are power laws with exponents −0.16 and −0.2, respectively.