Numerical Integration

The NumericalIntegrator class is the base class for all algorithms that implement numerical integration. It is discussed in the next section.

There are two main classes of algorithms. Fixed interval methods use the same integration rule over the entire interval. Iterations consist of dividing the interval into an ever greater number of subintervals until the estimated error is sufficiently small. The drawback of these methods is that they are relatively inefficient. The advantage is that they may give guaranteed upper or lower bounds for specific types of functions. For example, the trapezoid rule will consistently overestimate the integral of a convex function, while the midpoint rule will consistently underestimate the integral.

Adaptive methods divide intervals based on the estimated error of the integral. The intervals with the largest estimated error are subdivided first. This generally leads to fewer function evaluations.