Table 3 Results of using various ellipsoid surface samplers to generate \(10^8\) points on a triaxial ellipsoid, using two different random number generators. Each column shows (from left to right): algorithm name, run-time (in seconds), speed relative to naive rejection, acceptance rate, relative standard deviation (as a percentage of the mean), the \(\chi ^2\) statistic and whether \(\chi ^2\) is smaller than the critical value, \(\chi _\textrm{crit}^2 = 65\,033.6\)
From: Patch area and uniform sampling on the surface of any ellipsoid
Triaxial Ellipsoid: \((a,b,c) = (3,2,1)\) | ||||||
|---|---|---|---|---|---|---|
RNG | Lagged Fibonacci (lagfib4xor) | |||||
Algorithm | \(t_\textrm{run}\) (s) | speed | r | RSD (%) | \(\chi ^2\) | \(\chi ^2 < \chi _\textrm{crit}^2\) |
Naive Scale | 1.488 | 1.0 | 1.0 | 33.063 | 8716777.8 | No |
Grad Rej | 4.669 | 0.319 | 0.64832 | 3.222 | 63735.1 | Yes |
Grad (Trig) | 6.237 | 0.239 | 0.64838 | 3.263 | 64731.0 | Yes |
Grad (Pol) | 9.692 | 0.154 | 0.64830 | 3.27 | 64894.7 | Yes |
Area Rej | 5.956 | 0.25 | 0.71481 | 3.252 | 64109.0 | Yes |
Area (Pol) | 5.281 | 0.282 | 0.71489 | 3.234 | 64490.2 | Yes |
Area (Merc) | 15.64 | 0.095 | 0.20620 | 3.24 | 65870.9 | No |
Ray Method | 14.205 | 0.105 | 0.43220 | 3.269 | 64564.7 | Yes |
RNG | YARN5s (yarn5s) | |||||
|---|---|---|---|---|---|---|
Algorithm | \(t_\textrm{run}\) (s) | speed | r | RSD (%) | \(\chi ^2\) | \(\chi ^2 < \chi _\textrm{crit}^2\) |
Naive Scale | 3.137 | 1.0 | 1.0 | 33.087 | 8724015.4 | No |
Grad Rej | 8.592 | 0.365 | 0.64833 | 3.29 | 64737.6 | Yes |
Grad (Trig) | 10.03 | 0.313 | 0.64832 | 3.234 | 64219.2 | Yes |
Grad (Pol) | 13.074 | 0.24 | 0.64830 | 3.276 | 64459.8 | Yes |
Area Rej | 8.521 | 0.368 | 0.71483 | 3.243 | 64377.8 | Yes |
Area (Pol) | 8.217 | 0.382 | 0.71489 | 3.261 | 64086.4 | Yes |
Area (Merc) | 28.426 | 0.11 | 0.20619 | 3.227 | 65387.4 | No |
Ray Method | 21.785 | 0.144 | 0.43221 | 3.276 | 64486.1 | Yes |