Newton Fractals

Using Newton's Method to create Fractals by plotting convergence behavior on the complex plane. Functions used in this video include arctan(z), z^3-1, sin(z), z^8-15z^4+16. Example code and images available at https://github.com/osveliz/numerical-...

Correction: The derivative of arctan(x) should be 1/(1+x^2). This error is only impacts the slides, not the numerical examples nor fractals which used the correct derivative. A video covering this mistake can be found here    • Video Mistakes and How to Fix Them  

Chapters
0:00 Intro
0:16 Convergence Interval Recap
0:42 Imaginary Numbers
1:05 Newton's Method in Complex Plane
1:29 Basin of Convergence
1:38 Arctangent Fractal
1:50 Newton Fractal
2:10 Why Fractals Emerge
2:54 Example z^3-1
4:07 Example sin(z)
4:33 Example z^8+15z^4-16
4:50 Generalized Newton Fractal
5:05 Generalized Newton Fractal Examples
6:16 Summary
6:42 Thank You

Further Viewing:
Newton's Method    • Newton's Method  
Newton's Method Interval of Convergence    • Newton's Method Interval of Convergence  
Newton Bisection Hybrid (Newt-Safe)    • Newton Bisection Hybrid (Newt-Safe)  
Laguerre's Method and Fractal    • Laguerre's Method  
Halley's Method and Fractal    • Halley's Method  
Generalized Newton's Method    • Newton's Method for Systems of Nonlin...  

Further Reading:
Numerical Recipes http://numerical.recipes/
Wikipedia https://en.wikipedia.org/wiki/Newton_...

#Fractal #NewtonsMethod #NumericalAnalysis