This paper describes an estimation of the computational and memory complexities of Greengard-Rokhlin's Fast Multipole Algorithm (GRFMA). GRFMA takes a quad tree structure and six calculation processes. We consider a perfect a-ary tree structure and the number of floating-point operations for each calculation process. The estimation for both complexities shows that the perfect quad tree is the best and the perfect binary tree is the worst. When we apply GRFMA to the computation of realistic problems, volume scattering are the best case and surface scattering are the worst case. In the worst case, the computational and memory complexities of GRFMA are O(Llog2 L) and O(Llog L), respectively. The computational complexity of GRFMA is higher than that of the multilevel fast multipole algorithm.

The boundary element method (BEM), a representative method of numerical calculation of electromagnetic wave scattering, has been used for solving boundary integral equations. Using BEM, however, we finally have to solve a linear system of L equations expressed by dense coefficient matrix. The floating-point operation is O(L2) due to a matrix-vector product in iterative process. Greengard-Rokhlin's fast multipole algorithm (GRFMA) can reduce the operation to O(L). In this paper, we describe GRFMA and its floating-point operation theoretically. Moreover, we apply the fast Fourier transform to the calculation processes of GRFMA. In numerical examples, we show the experimental results for the computation time, the amount of used memory and the relative error of matrix-vector product expedited by GRFMA. We also discuss the convergence and the relative error of solution obtained by the BEM with GRFMA.