The Greengard-Rokhlin algorithm is a new and interesting method for computing long-range interactions in particle systems. Although the method already has been implemented and claimed to be superior to traditional and other methods, no reliable estimates of the size of the error of the method have been given. We illustrate what the error actually is for the two-dimensional case, and derive an estimate for it. The estimate has a simple analytic form which will allow its use in tuning the algorithm for best efficiency.

The fast multipole method is a new and interesting method for computing long range interactions in particle systems. Although the method already has been implemented and compared to traditional and other methods with respect to accuracy and speed, no accurate error estimates of the fast multipole method have been given. In this paper we develop an explicit though complicated form for the error in the three dimensional case, and derive an estimate for it. The estimate has a simple analytic form which will allow its use in tuning the method for best efficiency and for comparison of the method with other methods at the same accuracy.

The fast multipole method (FMM) has become an important alternative to traditional methods such as the Ewald method for computing the long-range interactions necessary to simulate charged or dipolar systems. In this paper, we present an improvement of this method, which we shall call the very fast multipole method (VFMM). The VFMM is shown to be a factor of about 1.2 faster than the FMM for two-dimensional systems and a factor about 2–3 times faster for three-dimensional systems without losing any accuracy for the worst case error.