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https://github.com/lanceberge/cuda-newton-fractals

Parallelize and visualize the Newton Iteration
https://github.com/lanceberge/cuda-newton-fractals

cpp cuda mathematical-modelling visualization

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Parallelize and visualize the Newton Iteration

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README 

        
# CUDA Newton Fractals



![order12](https://github.com/user-attachments/assets/d81ad4c3-ebb0-4d2b-89ec-c2b3adda2d4d)



Newton's Method is an iterative technique to find the roots of a polynomial. It begins with an initial guess of a root, then performs the iteration:



![iteration](fractals/iteration.svg)



until it converges to a root.



This project performs the iteration on initial guesses all across the complex plane. The iteration is performed asynchronously on each initial guess using CUDA.

Then, after these guesses have converged to the roots of the polynomial, they are color-coded based on which root they converged to.

This creates the famous fractal pattern: [Newton's Fractals](https://en.wikipedia.org/wiki/Newton_fractal).



This project also enables you to save snapshots of the evolution at each iteration, then to stitch them into a video:



![order12](https://github.com/user-attachments/assets/1c28f374-2c2b-4229-8ef1-98f93ea8819b)



## Running the Code



Example use:



```bash

# compile

make



# run code - this will output testName.png in fractals/

./bin/newton  [xPixels=?] [yPixels=?] [xRange=?] [yRange=?] [L1={true,false}] [step={true,false}]



## Or run my provided examples:

make runOrder7

make runOrder12

```



`testName` can be one of:



| Name          | Description                                       |

|--             |--                                                 |

| order7        | a given order 7 polynomial                        |

| order12       | a given order 12 polynomial - fractal shown above |

| anything else | you will be prompted to specify a polynomial      |



*Note*: If you use order7 or order12, xRange and yRange will already be set



| Parameter | Description                                                | Default |

|--         | --                                                         |--       |

| xPixels   | Number of horizontal pixels                                | 1000    |

| yPixels   | Number of vertical pixels                                  | 1000    |

| xyPixels  | Set both xPixels and yPixels                               | 1000    |

| xRange    | if 4, x values are between -4 and 4                        | 1.5     |

| yRange    | same as above but for the y values                         | 1.5     |

| xyRange   | set both xRange and yRange                                 | 1.5     |

| L1        | set to true if you want to use L1 norm to measure distance | false   |

| step      | set to true to output a png for each step                  | false   |



## Creating mp4s of the Evolution



You can also creat mp4s of the evolution of the fractals using:



```bash

make movie name=testName args="xyPixels=2000 xyRange=2"



## Or



# output pngs for each step

./bin/newton order12 step=true xyPixels=2000 xyRange=2



# then stitch into a movie

make stitchMovie name=order12

```



This will output [order12.mp4](fractals/order12.mp4) in [fractals](fractals).



## Producing Custom Fractals



You can also input your own polynomial, which will occur if `testName` isn't order7 or order12. You will be prompted to enter the roots of your polynomial, or 'random'. Example use:



```bash

$ ./bin/newton randomOrder20 xyPixels=2000 # higher resolution than defaults

Enter up to 99 characters of the roots of your polynomial separated by spaces:

ex. 5 4 3 2 1 to correspond to 5x^4 + 4x^3 + 3x^2 + 2x + 1

Or, enter 'random' to get a random polynomial

random

Enter [order] [max] [seed]

Order - the order of the polynomial

Max   - the max value of the coefficients (if 10, then all coefficients will be from -10 to 10

Seed  - seed the random polynomial (seeds drand48)

20 30 # an order 20 polynomial with roots between -30 and 30

```



This produces the fractal [fractals/randomOrder20.png](fractals/randomOrder20.png):



![randomOrder20](https://github.com/user-attachments/assets/9810616b-cd1d-46c2-8df8-f8715a79afc7)



If we produce an order 500 fractal:



![randomOrder500](https://github.com/user-attachments/assets/39ee02d0-1e68-4bc2-811b-b1af0c4a8604)



## Dependencies



| Dependency                                              | Command to install on Ubuntu             |

|--                                                       |--                                        |

| Libpng                                                  | sudo apt-get install libpng-dev          |

| CUDA and a cuda-capable GPU                             | sudo apt-get install nvidia-cuda-toolkit |

| ffmpeg (if you want to create a video of the evolution) | sudo apt-get install ffmpeg              |