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arXiv · 1907.03873

A fast simple algorithm for computing the potential of charges on a line

Zydrunas Gimbutas·Nicholas F. Marshall·Vladimir Rokhlin
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Abstract

We present a fast method for evaluating expressions of the form $$ u_j = \sum_{i = 1,i \not = j}^n \frac{α_i}{x_i - x_j}, \quad \text{for} \quad j = 1,\ldots,n, $$ where $α_i$ are real numbers, and $x_i$ are points in a compact interval of $\mathbb{R}$. This expression can be viewed as representing the electrostatic potential generated by charges on a line in $\mathbb{R}^3$. While fast algorithms for computing the electrostatic potential of general distributions of charges in $\mathbb{R}^3$ exist, in a number of situations in computational physics it is useful to have a simple and extremely fast method for evaluating the potential of charges on a line; we present such a method in this paper, and report numerical results for several examples.

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Zydrunas Gimbutas, Nicholas F. Marshall, Vladimir Rokhlin. 2019-11-25. A fast simple algorithm for computing the potential of charges on a line. https://doi.org/10.1016/j.acha.2020.06.002

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