I'm working on making a python library that shows how to code up some common quadrature methods. So far I've only made coded-up two methods, but hope to do more.
Simpson Quadrature is a commonly used method for numerically computing integrals. It is a good trade-off between simplicity and accuracy. Simpson Quadrature works by approximating the integrand as a sequence of quadratic segments.← Rhomberg Quadrature
Rhomberg Quadrature is one of the most accurate quadrature methods available, and is complicated to implement. It is used in applications where the integral must be known to a specified precision. Rhomberg Quadrature works by computing a sequence of approximations using the trapezoid method. Each approximation uses a a smaller grid spacing. Richardson extrapolation is then used to predict the limit as the grid spacing goes to zero.