A fast Gauss transform in one dimension using sum-of-exponentials approximations

Abstract

We present a fast Gauss transform in one dimension using nearly optimal sum-of-exponentials approximations of the Gaussian kernel. For up to about ten-digit accuracy, the approximations are obtained via best rational approximations of the exponential function on the negative real axis. As compared with existing fast Gauss transforms, the algorithm is straightforward for parallelization and very simple to implement, with only twenty-four lines of code in MATLAB. The most expensive part of the algorithm is on the evaluation of complex exponentials, leading to three to six complex exponentials FLOPs per point depending on the desired precision. The performance of the algorithm is illustrated via several numerical examples.