fractals from root-solving methods
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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Fractals from Root-Solving Methods
Daniel DreibelbisUniversity of North Florida
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OutlineDefine the problemExplore Newton’s Method, leading up to
Newton’s FractalsMess with Newton’s MethodTry this with other root-solving methods
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Root Solving
10 8 6 4 2 2
1 0
5
5
1 0
1 5
2 0
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Newton’s Method
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Newton’s Method
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Visualizing Newton’s Method
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Quadratic: Lame
z2 – 1 = 0
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Quadratic – Less Lame
z2 – 1 = 0
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Cubic – Not Lame
z3 – 1 = 0
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Cubic – Still not Lame
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Pretty Examples
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Pretty Examples
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Pretty Examples
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Pretty Examples
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Pretty Examples
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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.
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Why the fractal?
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Why Newton’s Method?
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Changing Newton’s Method
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Changing Newton’s Method
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Other Methods - Bisection
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Bisection on x3 – x = 0
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Other Methods - Secant
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Secant – Real Case
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Secant – Complex Case
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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
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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
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Steepest Descent
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Steepest Descent
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Steepest Descent
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Steepest Descent
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The End!Thanks!www.unf.edu/~ddreibel