Table 1 Results of using various sphere surface samplers to generate \(10^8\) points, using two different random number generators. Each column shows (from left to right): algorithm name, run-time (in seconds), speed relative to cubic rejection, acceptance rate, relative standard deviation (as a percentage), 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
Sphere: \((a,b,c) = (1,1,1)\) | ||||||
|---|---|---|---|---|---|---|
RNG | Lagged Fibonacci (lagfib4xor) | |||||
Algorithm | \(t_\textrm{run}\) (s) | speed | r | RSD (%) | \(\chi ^2\) | \(\chi ^2 < \chi _\textrm{crit}^2\) |
Cubic Rej | 2.803 | 1.0 | 0.52364 | 3.741 | 63834.1 | Yes |
Marsaglia | 1.205 | 2.326 | 0.78545 | 3.783 | 64958.6 | Yes |
Trig | 2.345 | 1.195 | 1.0 | 3.735 | 64346.7 | Yes |
Gaussian | 3.477 | 0.806 | 1.0 | 3.736 | 64059.1 | Yes |
Cook | 4.493 | 0.624 | 0.30844 | 3.725 | 64454.0 | Yes |
Area Rej | 5.513 | 0.508 | 0.63656 | 3.729 | 64260.0 | Yes |
RNG | YARN5s (yarn5s) | |||||
|---|---|---|---|---|---|---|
Algorithm | \(t_\textrm{run}\) (s) | speed | r | RSD (%) | \(\chi ^2\) | \(\chi ^2 < \chi _\textrm{crit}^2\) |
Cubic Rej | 7.226 | 1.0 | 0.52356 | 3.738 | 63920.5 | Yes |
Marsaglia | 2.999 | 2.409 | 0.78543 | 3.726 | 64931.0 | Yes |
Trig | 4.018 | 1.798 | 1.0 | 3.737 | 64792.5 | Yes |
Gaussian | 7.367 | 0.981 | 1.0 | 3.723 | 64339.9 | Yes |
Cook | 14.008 | 0.516 | 0.3084 | 3.769 | 64724.5 | Yes |
Area Rej | 8.027 | 0.9 | 0.63664 | 3.704 | 64468.3 | Yes |