An Introduction to Lattice Rules and their Generator Matrices

For the one-dimensional quadrature of a naturally periodic functionover its period, the trapezoidal rule is an excellent choice,its efficiency being predicted theoretically and confirmed inpractice. However, for s-dimensional quadrature over a hypercube,the s-dimensional product trapezoidal rule is not generallycost effective even for naturally periodic functions. The searchfor more effective rules has led first to number theoretic rulesand then more recently to lattice rules. This survey outlinesthe motivation for and present results of this theory. It isparticularly designed to introduce the reader to lattice rules.