Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes

Abstract

For a positive integer d, the d-dimensional Chebyshev-Frolov lattice is the Z-lattice in 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.

On generation and enumeration of orthogonal Chebyshev-Frolov lattices

“…Enumeration of CF-lattices in hypercubes has already been addressed, see e.g. [11,12,13,14]. For example, given a function f supported in the hypercube W ¼ ½À1=2; 1=2 d , one is interested in enumerating the quadrature nodes contributing to (3) with…”

“…In the recent paper [14], an optimal enumeration procedure is described, with optimality in the sense only the desired integers k are touched during the procedure. It is based on generating matrices A n that are more suitable for enumeration.…”

“…in his construction. Constructions based on Chebyshev polynomials were considered in [6,7,11,12,13,14] giving rise to the so called Chebyshev-Frolov lattices, i.e. fðPðx 1 Þ; .…”

“…We show in particular that using the appropriate reordering of rows and columns of the so-called Chebyshev-Vandermonde matrices, one is able to identify recurrences that are favorable to fast generation and enumeration of these lattices. We identify the polynomial sequence of Z½X which allows to derive the recurrences derived in [14] thus providing an explicit constructive approach. In § 4, we discuss the fast enumeration procedure of [14].…”

“…We identify the polynomial sequence of Z½X which allows to derive the recurrences derived in [14] thus providing an explicit constructive approach. In § 4, we discuss the fast enumeration procedure of [14].…”