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Table 1 The FMM kernels

From: A fast multipole method for stellar dynamics

Name Meaning Equation
P2P particle to particle (1)
P2M particle to multipole (3c)
M2M multipole to multipole (3d)
M2P multipole to particle (3g)
M2L multipole to local expansion (3b)
P2L particle to local expansion see table legend
L2L local expansion to local expansion (3e)
L2P local expansion to particle (3f)
  1. The tree code replaces direct summation (P2P) with P2M-M2M-M2P, while FMM uses P2M-M2M-M2L-L2L-L2P, see also Figure 2. The P2L kernel corresponds to equation (3b) with z A x a and M k l m a δ k 0 , i.e. F n m ( s B )= m a Θ n m ( s B x a ).