Basic Quadrature Rules
The quadrature rules generally have the form
where the nodes and weights are generally pre-computed.
In the simplest case, Newton–Cotes formulas of even degree are used, where the nodes are evenly spaced in the interval:
- .
When such rules are used, the points at which has been evaluated can be re-used upon recursion:
A similar strategy is used with Clenshaw–Curtis quadrature, where the nodes are chosen as
Or, when Fejér quadrature is used,
- .
Other quadrature rules, such as Gaussian quadrature or Gauss-Kronrod quadrature, may also be used.
An algorithm may elect to use different quadrature methods on different subintervals, for example using a high-order method only where the integrand is smooth.
Read more about this topic: Adaptive Quadrature
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