Fast solution of Foldy-Lax equations for two-dimensional radiation and scattering problems

The general fast multipole expressions of arbitrary order Hankel functions are derived by using lowering and raising operators of cylindrical harmonics. These expressions are then used to transform the dense matrix in Foldy-Lax equations into a combination of sparse matrices (aggregation, translation and disaggregation matrices). Thus, the computational complexity of such an algorithm is found to be of O(N-1.5) instead of O(N-2) of the traditional scattering matrix method, where N denotes the total harmonics number used to expand scattered of all the cylinders. The details of the implementation issues are investigated and, the accuracy and efficiency of this novel fast algorithm are verified by several numerical examples.