An Introduction to Lattice Rules and their Generator Matrices

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Abstract
For the one-dimensional quadrature of a naturally periodic function over its period, the trapezoidal rule is an excellent choice, its efficiency being predicted theoretically and confirmed in practice. However, for s-dimensional quadrature over a hypercube, the s-dimensional product trapezoidal rule is not generally cost effective even for naturally periodic functions. The search for more effective rules has led first to number theoretic rules and then more recently to lattice rules. This survey outlines the motivation for and present results of this theory. It is particularly designed to introduce the reader to lattice rules.

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