On the computational efficiencies of two FFT-based solution methods for the scalar multiple-scattering equation

We compare and improve the computational efficiencies of two FFT-based solution techniques for the scalar t-matrix multiple-scattering equation. The FFT operation requires field quantities be expressed on a regular grid and the two techniques differ in how to go about achieving this; the first technique uses the non-diagonal translation operator of the spherical multipole field, while the second method uses the diagonal translation operator of Rokhlin (1998). Because of its use of the non-diagonal operator, the first technique has been thought to require a greater number of FFT operations than the second one. We first show that both techniques require about the same number of FFT operations. After considering the symmetry relations of the non-diagonal and diagonal translation operators, we show that these symmetry relations can be used to reduce the respective memory requirements of the two FFT t-matrix methods