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# omgspace/pykdgrav

Fast N-body gravitational force and potential in Python Fetching latest commit…
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# Introduction

pykdgrav is a package that implements the Barnes-Hut method for computing the combined gravitational field and/or potential of N particles. We implement a kd-tree as a numba jitclass to achieve much higher peformance than the equivalent pure Python implementation, without writing a single line of C or Cython.

Despite the similar name, this project has no affiliation with the N-body code pkdgrav, however it is where I got the idea to use a kd-tree instead of an octree.

# Walkthrough

First let's import the stuff we want and generate some particle positions and masses

```import numpy as np
from pykdgrav import Accel, Potential
from pykdgrav.bruteforce import *```
```x = np.random.rand(10**5,3) # positions randomly sampled in the unit cube
m = np.random.rand(10**5) # masses```

Now let's compare the runtimes of the tree methods and brute force methods for computing the potential and acceleration. Note that all functions are jit-compiled by Numba, so the brute force in particular will run at C-like speeds.

```%time phi_tree = Potential(x,m)
%time a_tree = Accel(x,m)
%time phi_brute = BruteForcePotential(x,m)
%time a_brute = BruteForceAccel(x,m)```
``````CPU times: user 2.26 s, sys: 62 ms, total: 2.32 s
Wall time: 2.32 s
CPU times: user 3.94 s, sys: 66 ms, total: 4.01 s
Wall time: 4.01 s
CPU times: user 29.5 s, sys: 203 ms, total: 29.7 s
Wall time: 29.6 s
CPU times: user 1min 2s, sys: 274 ms, total: 1min 2s
Wall time: 1min 2s
``````

pykdgrav also supports OpenMP multithreading, but no support for higher parallelism is implemented nor planned. We can make it even faster by running in parallel (here on a dual-core laptop):

`%time a_tree = Accel(x,m,parallel=True)`
``````CPU times: user 5.38 s, sys: 83.8 ms, total: 5.46 s
Wall time: 2.18 s
``````

Nice, basically perfect scaling.

The treecode will almost always be faster than brute force for particle counts greater than ~10000. Below is a tougher benchmark for more realistic problem, run on a single core on my laptop. The particles were arranged in a Plummer distribution and an opening angle of 0.7 was used instead of the default 1: The method is approximate, using a Barnes-Hut opening angle of 1 by default; we can check the RMS force error here:

```delta_a = np.sum((a_brute-a_tree)**2,axis=1)
amag = np.sum(a_brute**2,axis=1)
print("RMS force error: %g"%np.sqrt(np.average(delta_a/amag)))```
``````RMS force error: 0.0345507
``````

We can improve the accuracy by choosing a smaller theta:

```a_tree = Accel(x,m,parallel=True, theta=0.7)
delta_a = np.sum((a_brute-a_tree)**2,axis=1)
amag = np.sum(a_brute**2,axis=1)
print("RMS force error: %g"%np.sqrt(np.average(delta_a/amag)))```
``````RMS force error: 0.0123373
``````

# Planned Features

• Greater OpenMP parallelism, e.g. in the tree-build algorithm.
• Support for computing approximate correlation and structure functions.
• Gravitational softening with abitrary kernels.

Stay tuned!

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