Given a function , compute (approximately) the definite integral
for some subset or to compute (approximately) the a derivatives , or etc., using only a finite (or small) number of function evaluations , .
The most efficient numerical integration methods have been developed for scalar problems (), including methods such as Gauss quadrature and Romberg integration. But there are good methods for many regions in higher dimensions. Regions with high dimensionality (e.g., ) are a particular challenge, as are integrands with singularities or localized "spikes".
Example 1: Integrate (or differentiate) the interpolant
If you have an effective and accurate method of interpolation, you can use that to construct an integration method as the integral of the interpolant.
Example 2: If you have a singularity that has an simple form, use product integration methods
For example, for computing integrals of the form
where is smooth on , determine a polynomial interpolant of , and compute
This technique is also useful for dealing with smooth functions with localized "spikes" that would otherwise require very many integration points, even with adaptive methods: for example, consider computing
with . By using an interpolant of on and computing the resulting integral exactly, the only errors incurred are due to the interpolation of , not the fact that the integral has a "spike".