Data Structures, Optimal Choice of Parameters, and Complexity Results for Generalized Multilevel Fast Multipole Methods in $d$ Dimensions

We present an overview of the Fast Multipole Method, explain the use of optimal data structures and present complexity results for the algorithm. We explain how octree structures and bit interleaving can be simply used to create efficient versions of the multipole algorithm in $d$ dimensions. We then present simulations that demonstrate various aspects of the algorithm, including optimal selection of the clustering parameter, the influence of the error bound on the complexity, and others. The use of these optimal parameters results in a many-fold speed-up of the FMM, and prove very useful in practice. UMIACS-TR-2003-2