Optimal quadrature points on the torus · n = 21
Place n equal-weight quadrature points on the torus T^d. Fix the kernel K(x,y) = Π (1 + 6·B₂({x_r − y_r})) with B₂(t) = t² − t + 1/6; the score is the squared worst-case error of the equal-weight rule, E = (1/n²)Σ K(x_i,x_j) − 1. Make it as small as possible.
Instanced = 3, n = 21
ObjectiveMinimize the worst-case integration error
Formal definition
- ContainerThe d-dimensional torus: coordinates modulo 1, written in [0, 1)
- SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
- ObjectiveMinimize the squared worst-case error of the equal-weight rule, E = (1/n²)Σ K(x_i,x_j) − 1; the kernel integrates to one, so E is non-negative
- KernelK = Π(1 + 6·B₂({x_r−y_r})); λ = 6 is this site's fixed kernel version, never to change — same family as the literature's periodic L2 discrepancy (λ = 3) and diaphony (λ = 2π²), deliberately its own parameter
- ScoringThe record is the exact integer n²S^{2d}·E with the denominator cleared; the page shows the error √E, rounded up at the twelfth decimal