# A kernel-independent treecode based on barycentric Lagrange interpolation

## TL;DR

This paper introduces a kernel-independent treecode that uses barycentric Lagrange interpolation to efficiently compute particle interactions, reducing complexity from quadratic to near-linear time, suitable for non-oscillatory kernels.

## Contribution

The paper presents a novel kernel-independent treecode based on barycentric Lagrange interpolation, enabling efficient and scalable particle interaction computations for non-oscillatory kernels.

## Key findings

- Reduces interaction computation from O(N^2) to O(N log N)
- Demonstrates effectiveness for 3D Stokeslet and rotlet systems
- Low memory usage allows easy parallelization

## Abstract

A kernel-independent treecode (KITC) is presented for fast summation of particle interactions. The method employs barycentric Lagrange interpolation at Chebyshev points to approximate well-separated particle-cluster interactions. The KITC requires only kernel evaluations, is suitable for non-oscillatory kernels, and it utilizes a scale-invariance property of barycentric Lagrange interpolation. For a given level of accuracy, the treecode reduces the operation count for pairwise interactions from $O(N^2)$ to $O(N\log N)$, where $N$ is the number of particles in the system. The algorithm is demonstrated for systems of regularized Stokeslets and rotlets in 3D, and numerical results show the treecode performance in terms of error, CPU time, and memory consumption. The KITC is a relatively simple algorithm with low memory consumption, and this enables a straightforward OpenMP parallelization.