Fractals from Root- Solving Methods Daniel Dreibelbis University of North Florida
Feb 24, 2016
Fractals from Root-Solving Methods
Daniel DreibelbisUniversity of North Florida
OutlineDefine the problemExplore Newton’s Method, leading up to
Newton’s FractalsMess with Newton’s MethodTry this with other root-solving methods
Root Solving
10 8 6 4 2 2
1 0
5
5
1 0
1 5
2 0
Newton’s Method
Newton’s Method
Visualizing Newton’s Method
Quadratic: Lame
z2 – 1 = 0
Quadratic – Less Lame
z2 – 1 = 0
Cubic – Not Lame
z3 – 1 = 0
Cubic – Still not Lame
Pretty Examples
Pretty Examples
Pretty Examples
Pretty Examples
Pretty Examples
Why the fractal?Near a critical point,
the tangent lines hit most of the x-axis. Thus most of the domain is mirrored near the critical point.
With two or more critical points, each critical point mirrors all of the other critical points.
Why the fractal?
Why Newton’s Method?
Changing Newton’s Method
Changing Newton’s Method
Other Methods - Bisection
Bisection on x3 – x = 0
Other Methods - Secant
Secant – Real Case
Secant – Complex Case
Other Methods – Steepest Descent f(x, y)=0 and g(x, y)=0 f(x, y)2 + g(x, y)2
2 1 0 1 2
2
1
0
1
2
Other Methods – Steepest Descent Re(z)=0 and Im(z)=0 Re(z)2 + Im(z)2
2 1 0 1 2
2
1
0
1
2
Steepest Descent
Steepest Descent
Steepest Descent
Steepest Descent
The End!Thanks!www.unf.edu/~ddreibel