# Fast multipole methods in three dimensions (FMM3D)¶

FMM3D is a set of libraries to compute N-body interactions governed by the Laplace and Helmholtz equations, to a specified precision, in three dimensions, on a multi-core shared-memory machine. The library is written in Fortran, and has wrappers for C, MATLAB, and Python. As an example, given $$M$$ arbitrary points $$y_j \in \mathbb{R}^{3}$$ with corresponding real numbers $$c_j$$, and $$N$$ arbitrary points $$x_{j} \in \mathbb{R}^{3}$$, the Laplace FMM evaluates the $$N$$ real numbers

(1)$u_{\ell} = \sum_{j=1}^M \frac{c_j}{\| x_{\ell} - y_{j}\|} ~, \qquad \mbox{ for } \; \ell=1,2,\ldots N ~.$

The $$y_j$$ can be interpreted as source locations, $$c_j$$ as charge strengths, and $$u_{\ell}$$ as the resulting potential at target location $$x_{\ell}$$.

Such N-body interactions are needed in many applications in science and engineering, including molecular dynamics, astrophysics, rheology, and the numerical solution of partial differential equations. The naive CPU effort to evaluate (1) is $$O(NM)$$. The FMM approximates (1) to a requested relative precision $$\epsilon$$ with linear effort $$O((M+N) \log^{3/2} (1/\epsilon))$$.

The FMM relies on compressing the interactions between well-separated clusters of source and target points at a hierarchy of scales using analytic outgoing, incoming, and plane-wave expansions of the interaction kernel and associated translation operators. This library is an improved version of the FMMLIB3D software, Copyright (C) 2010-2012: Leslie Greengard and Zydrunas Gimbutas, released under the BSD license. We provide two implementations of the library - an easy to install version with minimal dependencies, and a high-performance optimized version (which on some CPUs is 3x faster than the “easy” version). For detailed instructions, see installation. The major improvements are the following:

• The use of plane wave expansions for diagonalizing the outgoing to incoming translation operators
• Vectorization of the FMM, to apply the same kernel with the same source and target locations to multiple strength vectors
• Optimized direct evaluation of the kernels using the SCTL library
• A redesign of the adaptive tree data structure

For sources and targets distributed in the volume, this code is about 4 times faster than the previous generation on a single CPU core, and for sources and targets distributed on a surface, this code is about 2 times faster.

Note

The present version of the code does not incorporate high frequency versions of the FMM. The plane wave expansions are used throughout for the Laplace FMM and for the Helmholtz case in the low frequency regime (for box sizes up to 32 wavelengths). At higher frequencies, the Helmholtz FMM uses the same point and shoot translation operators as FMMLIB3D, which results in sub-optimal performance.

Note

For very small repeated problems (less than 1000 input and output points), users should also consider dense matrix-matrix multiplication using BLAS3 (eg DGEMM,ZGEMM).