Error in formula for chiN

[cgd8d]

cmaes.c uses the formula (https://github.com/CMA-ES/c-cmaes/blob/0be438f8276cad559a4c220b7f683e23874605db/src/cmaes.c#L313):

t->chiN = sqrt((double) N) * (1. – 1./(4.*N) + 1./(21.*N*N));

The number “21” is also used in your tutorial at https://www.lri.fr/~hansen/cmatutorial.pdf. However, I believe that it is not the correct asymptotic expansion of chiN. See http://www.wolframalpha.com/input/?i=series+expansion+for+sqrt%282%29*gamma%28%28n%2B1%29%2F2%29%2Fgamma%28n%2F2%29+at+n%3Dinfinity, which indicates that “21” should be changed to “32”.

Alternatively, C99 standardizes the tgamma function, so if you assume C99 compilers then it could be replaced entirely with

t->chiN = sqrt(2.) * tgamma((double)(N+1)/2)/tgamma((double)N/2);

or, for to avoid overflow with roughly N > 340, the slightly less accurate

t->chiN = sqrt(2.) * exp(lgamma((double)(N+1)/2) – lgamma((double)N/2));

[nikohansen]

which indicates that “21” should be changed to “32”

I don't think so, as Wolfram doesn't give the coefficient for the quadratic term. I am also not sure that it is sufficient to consider the expansion for n=\infty only. On the other hand, I am almost certain that the formula used in the code is triple-checked to be accurate (enough) in particular also for any given finite dimension.

[cgd8d]

Your expression has a leading sqrt(N), so both you and Wolfram provide n^0.5, n^-0.5, and n^-1.5 terms. But you are right: for N <= 4, your expression is more accurate: http://www.wolframalpha.com/input/?i=plot+abs%28sqrt%28n%29*%281-1%2F%284*n%29%2B1%2F%2821*n%5E2%29%29+-+sqrt%282%29*gamma%28%28n%2B1%29%2F2%29%2Fgamma%28n%2F2%29%29%2C+abs%28sqrt%28n%29*%281-1%2F%284*n%29%2B1%2F%2832*n%5E2%29%29-sqrt%282%29*gamma%28%28n%2B1%29%2F2%29%2Fgamma%28n%2F2%29%29%2C+n%3D0+to+10.

So, fair point. I assumed it was a straightforward expansion, but clearly some extra thought went into it which I missed.