ABSTRACT
A typical use of the FMM method involves systems containing millions or billions of particles. For systems of this size, the cost of the computational time for evaluating particle – particle interactions becomes problematic. Despite the FMM method reducing the cost so that it grows only linearly with system size, the computational effort is still considerable. The need to speed up the conventional FMM method motivates the search for its more efficient formulations. One option involves the replacement of the conventional formulation of multipole translations, which scales as the fourth power of the size of the truncated multipole expansion, by translation along the z-axis to reduce the scaling to cubic. This chapter describes the derivation of equations for the M2M, M2L, and L2L multipole operations defined in terms of translation along the z-axis and provides example of their naïve and optimized computer implementations. Optimization includes avoiding the use of conditional statements and of the direct computation of factorials in the program code.
