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A Fast Summation Method for Oscillatory Lattice Sums



Ryan Denlinger, Leslie Greengard, Zydrunas Gimbutas, Vladimir Rokhlin


We present a fast summation method for lattice sums of the type which arise when solving wave scattering problems with periodic boundary conditions. While there are a variety of effective algorithms in the literature for such calculations, the approach presented here is new and leads to a rigorous analysis of Wood's anomalies. These arise when illuminating a grating at specific combinations of the angle of incidence and the frequency of the wave, for which the lattice sums diverge. They were discovered by Wood in 1902 as singularities in the spectral response. The primary tools in our approach are the Euler-Maclaurin formula and a steepest descent argument. The resulting algorithm has super-algebraic convergence and requires only milliseconds of CPU time.
Journal of Mathematical Physics


Lattice sums, Euler-Maclaurin formula, Wood’s anomalies


Denlinger, R. , Greengard, L. , Gimbutas, Z. and Rokhlin, V. (2017), A Fast Summation Method for Oscillatory Lattice Sums, Journal of Mathematical Physics, [online],, (Accessed April 24, 2024)
Created March 5, 2017, Updated October 12, 2021