Numerical integration, quadrature

A simple procedure for calculating a one-dimensional integral

is to divide the interval [a,b] into smaller intervals [u,u+h], where File:Hepa img777.gif and then use approximations like Simpson's rule,

or three-point Gauss integration

Errors for the complete integral are O(h⁴) and O(h⁶), respectively, and the two methods are said to be of order 4 and 6. Note that these error estimates are invalid if the integrand has singularities or discontinuities. For more detail, see Wong92.