Table 2 Results of using various ellipsoid surface samplers to generate \(10^8\) points on an oblate spheroid, 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
Oblate Spheroid: \((a,b,c) = (3,3,1.5)\) | ||||||
|---|---|---|---|---|---|---|
RNG | Lagged Fibonacci (lagfib4xor) | |||||
Algorithm | \(t_\textrm{run}\) (s) | speed | r | RSD (%) | \(\chi ^2\) | \(\chi ^2 < \chi _\textrm{crit}^2\) |
Naive Scale | 1.493 | 1.0 | 1.0 | 24.878 | 5005750.1 | No |
Grad Rej | 4.284 | 0.349 | 0.69007 | 3.316 | 64726.9 | Yes |
Grad (Trig) | 5.83 | 0.256 | 0.69009 | 3.336 | 64634.3 | Yes |
Grad (Pol) | 9.447 | 0.158 | 0.69014 | 3.318 | 64978.8 | Yes |
Area Rej | 4.995 | 0.299 | 0.76086 | 3.307 | 64426.4 | Yes |
Area (Pol) | 3.223 | 0.463 | 0.76099 | 3.303 | 64032.3 | Yes |
Area (Merc) | 10.794 | 0.138 | 0.21841 | 3.461 | 73037.1 | No |
Ray Method | 9.281 | 0.161 | 0.69011 | 3.309 | 64192.8 | Yes |
RNG | YARN5s (yarn5s) | |||||
|---|---|---|---|---|---|---|
Algorithm | \(t_\textrm{run}\) (s) | speed | r | RSD (%) | \(\chi ^2\) | \(\chi ^2 < \chi _\textrm{crit}^2\) |
Naive Scale | 3.155 | 1.0 | 1.0 | 24.87 | 5006451.3 | No |
Grad Rej | 8.02 | 0.393 | 0.69001 | 3.291 | 64402.6 | Yes |
Grad (Trig) | 9.331 | 0.338 | 0.69010 | 3.346 | 64716.3 | Yes |
Grad (Pol) | 12.433 | 0.254 | 0.69012 | 3.283 | 64401.4 | Yes |
Area Rej | 7.286 | 0.433 | 0.76088 | 3.295 | 63825.3 | Yes |
Area (Pol) | 6.285 | 0.502 | 0.76093 | 3.317 | 63805.3 | Yes |
Area (Merc) | 17.333 | 0.182 | 0.21846 | 3.39 | 73010.7 | No |
Ray Method | 14.121 | 0.223 | 0.69002 | 3.318 | 64134.3 | Yes |