Novel calculation rule in multipole algorithm

The fast multipole algorithm (FMA) and the multilevel fast multipole algorithm (MLFMA) are ways to expedite matrix-vector multiplication in the CG method and are researched widely as a calculation method in electromagnetic scattering analysis. In a simple 2D MoM case, the Green's operator is factorized using Bessel and Hankel functions with the assistance of the addition theorem as the non-diagonal form, and this expression can be rewritten in the Fourier space as the diagonal form. Even though the Hankel function series has the divergence property, the summation of the non-diagonal form converges because the Bessel function diminishes more rapidly. On the other hand, the diagonal form diverges for a large truncation number. Even if the most optimum truncation number is selected in the diagonal form, the error does not become extremely small, especially in the small buffer case. This paper proposes a novel calculation rule that uses two equations, the diagonal form and the non-diagonal. By switching between these two expressions, the error coming from the nearest neighbor interactions can be suitably suppressed to the smaller level and the calculation can be realized with high precision accuracy. Not only minimizing the error but also realizing the required error level is also possible in this new rule. The relations between several parameters, such as truncation number, buffer number and box size, are demonstrated.