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

Abstract

We present a fast method for evaluating expressions of the form $$u_j=\sum _{i=1,i\ne j}^n\frac{{\alpha }_i}{x_i-x_j},\text{for}j=1,\dots ,n,$$ where ${\alpha }_i$ 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.

16 pages

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