The Role of Structured Matrices for the Construction of Integration Lattices

Date of Online Publication: 22/12/2006
Keywords: multivariate numerical integration, quadrature, cubature, lattice rules
Authors: R. Cools, D. Nuyens
Pages: 257-272
An overview is presented of the role played by structured matrices in the construction of lattice rules for numerical integration. Two methods which make use of (skew-)circulant matrices are discussed. The first method is for constructing lattice rules which are exact for trigonometric polynomials up to a certain degree, while the second method optimizes for the worst-case error in some function space. By using these structured matrices, both methods deliver lattice rules which would otherwise be impossible to construct. We also present some practical advice on how to use a lattice rule as a sequence and obtain an estimate for the integration error. We present a small algorithm to generate the points from any lattice as a sequence and demonstrate this technique on a numerical example. It can be observed that this method is a handy tool for anybody wanting to use a lattice rule without predetermining the number of points to be used.

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