Numerical Integration

Numerical integration is the numerical approximation of the integral of a function. For a function of one variable, it amounts to finding the area under the graph of the function. That is finding I where

For a function of two variables it is equivalent to finding an approximation to the volume under the surface. Numerical integration is often also referred to as quadrature or sometimes cubature for functions of two or more variables. Returning to the one variable case, numerical integration involves finding the approximation to an integral of a function f(x) through its evaluation at a set of discrete points. There are two distinct approaches to this. Firstly methods like the trapezium rule or Simpson's rule determine the integral through evaluating f(x) at regularly spaced points. These are generally referred to as Newton-Cotes formulae.

Alternative methods termed Gaussian Quadrature methods have arisen that select irregularly-placed evaluation points, chosen to determine the integral as accurately as possible with a given set of points.

Gaussian Quadrature methods are important as they often lead to very efficient methods. In numerical integration the efficiency of the method relates to the accuracy obtained with respect to the number of evaluations of the function f(x). In intensive methods such as the boundary element method integrations may need to be performed millions of times so the efficiency of the methods needs to be considered sometimes.

In general, care must be taken to match the numerical integration method to the expected nature of the function f(x). For example it may be known that f(x) is regular. On the other hand f(x) may be singular or oscillatory and will then need special treatment. Often a special method called a product integration method can be developed for the integration of functions of the form f(x) = w(x)g(x) where w(x) is a pre-set function and the function and g(x) is known to be a relatively nice function.

There are books devoted to numerical integration. Numerical integration is a basic numerical method and the subject is generally covered in books on numerical methods or numerical analysis (see below). Numerical methods for carrying out numerical integration can often be easily programmed and can also be found in general numerical libraries.