Error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem

The numerical solution of wave scattering from large objects or from a large cluster of scatterers requires excessive computational resources and it becomes necessary to use approximate-but fast-methods such as the fast multipole method; however, since these methods are only approximate, it is important to have an estimate for the error introduced in such calculations. An analysis of the error for the fast multipole method is presented and estimates for truncation and numerical integration errors are obtained. The error caused by polynomial interpolation in a multilevel fast multipole algorithm is also analyzed. The total error introduced in a multilevel implementation is also investigated numerically.