What is quadrature?

The basic idea of quadrature is to obtain the numerical approximation for the integral of the function \(f(x)\) over the domain \(\mathcal{D}\) in terms of a weighted sum over discrete points,

\[\int_{\mathcal{D}} f(x) \mathrm{d}x \approx \sum_{i=1}^n f(x_i) w_i \;.\]

Different approaches exist to determine the optimal number and distribution of sample points, \(x_i\), and their corresponding weights, \(w_i\), for a given domain. This gives rise to a variety of classes of quadrature methods. See our advanced guide for a brief overview of different methods.

Based on Fubini’s theorem, lower-dimensional quadratures can be combined to construct tensor product rules for muti-dimensional integrals.