We present a matrix interpretation of the three-dimensional fast multipole
method (FMM). The FMM is for efficient computation of gravitational/electro
static potentials and fields. It has found various applications and inspire
d the design of many efficient algorithms. The one-dimensional FMM is well
interpreted in terms of matrix computations. The three-dimensional matrix v
ersion reveals the underlying matrix structures: and computational techniqu
es used in FMM. It also provides a unified view of algorithm variants as we
ll as existing and emerging implementation of the FMM.