A generalized fast multipole method for nonoscillatory kernels

We present a modi cation of the fast multipole method (FMM) in two dimensions. While previous implementations of the FMM have been designed for harmonic kernels, our algorithm works for a large class of kernels that satisfy fairly general conditions, amounting to the kernel being sufficiently smooth away from the diagonal. Our algorithm approximates appropriately chosen parts of the kernel with tensor products of Legendre expansions and uses the singular value decomposition (SVD) to compress the resulting representations. The obtained singular function expansions replace the Taylor and Laurent expansions used in the original FMM. The algorithm requires O(N) operations and is stable and robust. The performance of the algorithm is illustrated with numerical examples.