Error Analysis of the GRAPE-1 Special-Purpose N-Body Machine

We discuss the numerical accuracy of GRAPE-1, a special-purpose computer for astrophysical N-body calculations. The word length is the primary factor that determines the construction cost of such a special-purpose computer. Regarding GRAPE-1, we adopted the minimal word length with which we can obtain physically correct results. We used 16-bit fixed-point numbers for the positions of particles, 48-bit fixed-point numbers for the accumulation of force, and 8-bit floating-point numbers for all other force calculations. The theoretical accuracy of the force calculated by GRAPE-1 is ~ 0.4% for a homogeneous sphere with a softening parameter of the order of the average interparticle distance, independent of the number of particles, except for very small N (such as N <~ 100). The integration error for a typical particle caused by this round-off error scales as a two-body relaxation effect, and is roughly 10--100 times smaller than the relaxation effect itself. This implies that the numerical accuracy of GRAPE-1 is sufficient for collisionless calculations, since two-body relaxation dominates the numerical error of collisionless N-body calculations. The results of calculations on the GRAPE-1 system support our theoretical estimate.