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