Rotating around the quartic angular momentum barrier in fast multipole method calculations

The fast multipole method (FMM) evaluates the potential interactions of point charges (or masses) in time scaling linearly with their number. The FMM requires the ability to manipulate multipole and Taylor expansions. These manipulations are conventionally performed using a set of O(L4) translation operators, where L is the degree of the expansion. We introduce a method which achieves O(L3) scaling by rotating the expansions such that translation is always along the quantization or Z axis. It retains the FMM error bound. An overall speedup approaching a factor of 3 for relatively high orders of multipoles (L=21) is demonstrated. This procedure makes high accuracy potential evaluations substantially more efficient.