Fractional tiers in fast multipole method calculations

One defining characteristic of the fast multipole calculation is the number of tiers (depth of tree) used to group the particles. For three dimensions, the standard boxing scheme restricts the number of lowest level boxes to be a power of eight. We present a method which through a simple scaling of the particle coordinates allows an arbitrary number of lowest level boxes. Consequently, one can better balance the near-field and far-field work by minimizing the variation in the number of particles per lowest level box from its optimal value. Test calculations show systems where this method gives a speedup approaching two times.