A class of methods called lattice methods for numerical evaluation of multiple integrals is described. These lattice methods are useful for integration in dimensions much greater than 1 (6-20 or even more), which occurs e.g. in atomic physics, quantum chemistry, and statistical mechanics.
A brief tour of lattice rules is carried out. A mathematical classification of lattice rules which is useful for computation is accomplished. The method of good lattice points is explained. The averaging technique may be used for more general lattice rules. The error is estimated by randomization.
The question of how good the lattice rules of higher rank may be is studied from the structure of lattice rules (they are represented as multiple sums.) Lattice rules of maximal rank have attractive error bounds. Certain lattice rules of intermediate rank for nonperiodic integrands are considered. An expression is found for the number of distinct lattice rules of a given order. The practical implementation of lattice rules is discussed. Embedded sequences of quadrature rules open the possibility of obtaining an error estimate.