We present a matrix interpretation of the three-dimensional fast multipole method (FMM). The FMM is for efficient computation of gravitational/electrostatic potentials and fields. It has found various applications and inspired the design of many efficient algorithms. The one-dimensional FMM is well interpreted in terms of matrix computations. The three-dimensional matrix version reveals the underlying matrix structures and computational techniques used in FMM. It also provides a unified view of algorithm variants as well as existing and emerging implementations of the FMM.
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Sun et al. (2001) studied this question.
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