Numerical integration or quadrature approximates an integral
over a fixed interval. The
Extreme Optimization Numerical Libraries
contain a variety of algorithms that are
can be useful in varying situations.
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.