Error analysis for the truncation of multipole expansion of vector Green's functions

The advent of fast algorithms in computational electromagnetics has permitted the solution of integral equations with an unprecedented number of unknowns. This is the consequence of the development of the fast multipole algorithms (FMA) and the multilevel fast multipole algorithms (MLFMA). Such algorithms allow a matrix-vector multiplication to be performed in O(NlogN) operations or less for many scattering problems. Moreover, the memory requirements of these methods are O(NlogN), or almost matrix free. Using the fast matrix-vector multiplications in an iterative solver, problems for integral equations involving millions of unknowns have been solved previously. One of the most important mathematical formulas in FMA is the addition theorem. In the numerical implementation of the addition theorem, the infinite series should be truncated. The error analysis for the truncation error in the scalar Green's functions has been done by many researchers. In this paper, the error analysis for the truncation error in the multipole expansion of vector Green's functions is given.