Numerical integration of functions with singularities

The numerical integration of functions with singularities, at which the function itself or some of its derivatives become infinite is discussed. The integration interval can be divided into subintervals that has a single singularity on each of them that lies at a subinterval endpoint. Aitken's method, which can also be made recursive, can be extended to integrals over unbounded domains and to other problems. Aitken proposed the extrapolation procedure to determine an approximate limit of the geometric progression from its partial sums. This procedure, when applied to numerical integration, give a considerable improvement in the accuracy. Aitken's recursive refinements approach polynomial dependencies and reduce the error by many orders of magnitude. Aitken's method improves the accuracy without increasing the amount of computations, and can be also used to solve other singularity involving problems by grid methods.