Fast multipole method

Imagine you have a lot of objects that you want to understand how they interact with each other. One way to do this is to calculate the force between every pair of objects. But what if you have millions or billions of objects? This would take a very long time, even with a powerful computer.

The fast multipole method is a way to speed up this computation. It's kind of like making a shortcut so you don't have to do all the work.

First, you group the objects together into clusters that are far away from each other. Then, you calculate the force between the clusters instead of the individual objects. This reduces the number of calculations you need to do.

But wait, there's more! You can also use a trick called multipole expansion. This means that instead of calculating the force between each object, you use a formula to estimate the force based on the position and characteristics of the clusters.

This method is really fast because you're doing fewer calculations and using estimates instead of exact values. But it's also pretty accurate because you're grouping the objects together in a smart way.

So imagine you're building a big puzzle with a lot of pieces. Instead of trying to put all the pieces together individually, you group them into smaller clusters and work on the clusters first. This saves you time and effort, but you still get a complete picture of the finished puzzle. That's what the fast multipole method does!