An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules

Abstract

Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in R(s) and are generalizations of 'number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962)-themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variable, these bounds are conveniently characterized by the figure of merit rho, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order N (that is, having N nodes) then becomes that of finding rules with large values of rho. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.