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Fractals from Root-Solving Methods. Daniel Dreibelbis University of North Florida. Outline. Define the problem Explore Newton’s Method, leading up to Newton’s Fractals Mess with Newton’s Method Try this with other root-solving methods. Root Solving. Newton’s Method. Newton’s Method.
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Fractals from Root-Solving Methods Daniel Dreibelbis University of North Florida
Outline • Define the problem • Explore Newton’s Method, leading up to Newton’s Fractals • Mess with Newton’s Method • Try this with other root-solving methods
Quadratic: Lame z2 – 1 = 0
Quadratic – Less Lame z2 – 1 = 0
Cubic – Not Lame z3– 1 = 0
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.
Other Methods – Steepest Descent f(x, y)=0 and g(x, y)=0 f(x, y)2 + g(x, y)2
Other Methods – Steepest Descent Re(z)=0 and Im(z)=0 Re(z)2 + Im(z)2
The End! • Thanks! • www.unf.edu/~ddreibel