On the scaling of multipole methods for particle–particle interactions

There is some controversy regarding the scaling of the fast multipole method (FMM). It has recently been proven by Aluru that the FMM is not a linear scaling method, but an O(N log4 N) method. Aluruʹs proof cannot be applied to typical computational chemistry calculations where the required precision is smaller than the machine accuracy. In this Letter, we deal with this kind of situation and give a rigorous bound to the scaling and a statistical estimate. We also perform numerical tests. Our results agree with Aluruʹs proof. The scaling of other methods that use multipoles is also discussed.