# Project: Fast Multipole Methods

The purpose of this project is to write the tree code and fast multipole methods for the N-body summation problem.

Reference:

## Contents

## Step 1: Direct Sum

Generate two random vectors x, y with length N. Although the direct sum can be implemented in the double for loops, in MATLAB, it is better to generate the matrix first and then compute the matrix-vector product for another random vector q.

## Step 2: Compute the weight

To store the weight in different level, use `cell` structure. Use a for loop of i=1:N to compute the weight first and then try to remove this for loop to speed up your code. The loop over levels is small (only logN times) and thus can be kept.

## Step 3: Evaluation

First write code to find the interaction list. Then loop over each cell in a given level and compute the far field in the interaction list. In the fines level, add the near field by direct sum or matrix-vector product using a small matrix.

## Step 4: Test

- Chose N small and J = 1. Make sure the code works for one level (only four intervals) first by comparing the result using tree algorithm with the result in Step 1.
- Test the performance for different N and plot the CPU time vs N for both direct method and tree code.

## Step 5 (optional): Fast Multipole Methods

Modify the tree code to fast multipole methods.

- Compute the weight by restriction from the fine grid to coarse grid.
- Implement the M2L: multipole expansion to local expansion
- Change the evaluation of far field in the interaction list to the merge of coefficients b in the local expansion.
- Translate the local expansion using the prolongation operator.
- Evaluate in the finest level.
- Plot the CPU time vs N to confirm the O(N) complexity.