Adaptive quadrature

In applied mathematics, adaptive quadrature is a process in which the integral of a function ${\displaystyle f(x)}$ is approximated using static quadrature rules on adaptively refined subintervals of the integration domain. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well behaved" integrands, but are also effective for "badly behaved" integrands for which traditional algorithms fail.

Adaptive quadrature follows the general scheme

An approximation ${\displaystyle Q}$ to the integral of ${\displaystyle f(x)}$ over the interval ${\displaystyle [a,b]}$ is computed (line 2), as well as an error estimate ${\displaystyle \varepsilon }$ (line 3). If the estimated error is larger than the required tolerance ${\displaystyle \tau }$(line 4), the interval is subdivided (line 5) and the quadrature is applied on both halves separately (line 6). Either the initial estimate or the sum of the recursively computed halves is returned (line 7).

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