Numerical integration, quadrature

A simple procedure for calculating a one-dimensional integral

File:Hepa img776.gif

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,

File:Hepa img778.gif

or three-point Gauss integration

File:Hepa img779.gif

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.

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