fmmgen: Automatic Code Generation of Operators for Cartesian Fast Multipole and Barnes-Hut Methods

The Barnes-Hut and Fast Multipole Methods are widely utilised methods applied in order to reduce the computational cost of evaluating long range forces in $N$-body simulations. Despite this, applying existing libraries to simple problems with higher order source points, such as dipoles, is not straightforward or efficient because individual libraries are optimised towards specific problems, normally solving for the potential and field of a set of Coulombic particles. In this paper we detail the implementation and testing of a software package, fmmgen, in which the source code for Barnes-Hut and Fast Multipole operator functions for calculating calculate the potential, field or both from arbitrary ordered sources is easily generated through symbolic algebra.