Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes
Kosuke Suzuki, Takehito Yoshiki
TL;DR
The paper develops a high-performance algorithm for enumerating Chebyshev-Frolov lattice points in axis-parallel boxes for dimensions $d=2^n$ up to 32. It exploits a recursive generating-matrix structure $A_n$ derived from the roots of the Chebyshev polynomial, reducing $2^n$-dimensional enumeration to lower dimensions and eventually to 1D steps, implemented via recursive and tail-recursive schemes. This enables efficient use of Chebyshev-Frolov lattices as nodes in Frolov's cubature formula, including its randomized variant, by mapping lattice points to cubature nodes through scaled generators $A_{n,N}$. The results demonstrate practical enumeration up to $d=32$, with clear guidance on when the method is most effective and considerations for numerical accuracy. This advances the applicability of optimal cubature in higher dimensions by providing a scalable, exact node enumeration method.
Abstract
For a positive integer $d$, the $d$-dimensional Chebyshev-Frolov lattice is the $\mathbb{Z}$-lattice in $\mathbb{R}^d$ generated by the Vandermonde matrix associated to the roots of the $d$-dimensional Chebyshev polynomial. It is important to enumerate the points from the Chebyshev-Frolov lattices in axis-parallel boxes when $d = 2^n$ for a non-negative integer $n$, since the points are used for the nodes of Frolov's cubature formula, which achieves the optimal rate of convergence for many spaces of functions with bounded mixed derivatives and compact support. The existing enumeration algorithm for such points by Kacwin, Oettershagen and Ullrich is efficient up to dimension $d=16$. In this paper we suggest a new enumeration algorithm of such points for $d=2^n$, efficient up to $d=32$.