By iterative algorithm, I mean a numerical algorithm that works by improving on a previous approximation to obtain a more accurate approximation. An example is Newton's Method.
The numerical integration algorithms I know (e.g. Simpson's rule, Trapezoidal rule, etc.) do not work like this. The exception, perhaps, being Monte-Carlo. Is there a way one can tweak something like Simpson's rule to make it iterative?
One way to improve a numerical integration is to refine the mesh of sample points. Using a number of sample points that is a power of $2$ makes this easier. The idea is to compute the integral using $n$ points and then with $2n$ points (reusing the work done for the $n$ original points). If the two values are close, then you're done. Otherwise, keep on refining.
A variant of this idea is an adaptive method. See for instance the Adaptive Simpson's method.
Trapezoidal Rule, Simpson's Rule do kind of work like this. Once we have TRAP for $n$, we can get TRAP for $2n$ by recycling TRAP for $n$ and using the function values at $n$ additional points.
For something closer to iterative, please see Richardson extrapolation, and more particularly Romberg integration.
