From the summation-by-parts algorithm to pi

The summation-by-parts algorithm has been introduced as a simple way of accelerating the convergence of some series arising in the modal evaluation of Green's functions in waveguides and cavities. It can be considered to be a discrete equivalent of the classical integration-by-parts technique, whence the name. An evaluation of the real improvement that this procedure introduces in electromagnetic problems is not easy, and is frequently obscured by many other factors intervening in the same problem. In this short paper, a simpler way of formulating the algorithm is first introduced. Then, the performance of the summation-by-parts procedure is clearly stated by applying it to one of the most basic problems in numerical mathematics: the evaluation of the transcendental number, Pi