The Fast Gauss Transform

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Abstract
Many problems in applied mathematics require the evaluation of the sum of N Gaussians at M points in space. The work required for direct evaluation grows like N M as N and M increase; this makes it very expensive to carry out such calculations on a large scale. In this paper, an algorithm is presented which evaluates the sum of N Gaussians at M arbitrarily distributed points in C . (N + M) work, where C depends only on the precision required. When N = M = 100,000, the algorithm presented here is several thousand times faster than direct evaluation. It is based on a divide-and-conquer strategy, combined with the manipulation of Hermite expansions and Taylor series.

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