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

## TL;DR

This paper introduces a rapid and straightforward algorithm to efficiently compute electrostatic potentials generated by charges positioned on a line, significantly simplifying calculations in relevant physics applications.

## Contribution

The paper presents a novel, simple, and fast algorithm specifically designed for calculating potentials of line charges, filling a gap in computational physics methods.

## Key findings

- Algorithm significantly reduces computation time.
- Numerical tests demonstrate high accuracy.
- Applicable to various charge distributions on a line.

## Abstract

We present a fast method for evaluating expressions of the form $$ u_j = \sum_{i = 1,i \not = j}^n \frac{\alpha_i}{x_i - x_j}, \quad \text{for} \quad j = 1,\ldots,n, $$ 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.