Fractional Part - A Difficult Integral

[ttw]

Numerical integration of things like Frac(x)dx do not gain by using high-order rules (Simpson's, Gauss, etc.) because the first derivative is not continuous.

Monte Carlo (and quasi Monte Carlo) does no worse than its usual performance (which isn't that good anyway) as the variance (or variation) is small.

One QMC example is to use the points (2j-1)/2N for j=1,N as an N-point integration formula. Another is to use Frac(N*Sqrt(2)) as a sequence to integrate these types of functions.

The last sequence is easy to compute; just add Frac(Sqrt(2)) at each step and reduce below 1.