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Fractals from Root-Solving Methods

This presentation from Daniel Dreibelbis at the University of North Florida delves into the intricate world of root-solving methods, particularly focusing on Newton's Method and its fascinating fractals. Starting with problem definition, the exploration leads into the mechanics of Newton’s Method and extends to visualizations, demonstrating both quadratic and cubic equations. The discussion also includes other root-solving techniques like bisection and secant methods, examining their interaction near critical points and the resulting fractal patterns. Join us for a journey into visual mathematics!

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Fractals from Root-Solving Methods

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  1. Fractals from Root-Solving Methods Daniel Dreibelbis University of North Florida

  2. 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

  3. Root Solving

  4. Newton’s Method

  5. Newton’s Method

  6. Visualizing Newton’s Method

  7. Quadratic: Lame z2 – 1 = 0

  8. Quadratic – Less Lame z2 – 1 = 0

  9. Cubic – Not Lame z3– 1 = 0

  10. Cubic – Still not Lame

  11. Pretty Examples

  12. Pretty Examples

  13. Pretty Examples

  14. Pretty Examples

  15. Pretty Examples

  16. 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.

  17. Why the fractal?

  18. Why Newton’s Method?

  19. Changing Newton’s Method

  20. Changing Newton’s Method

  21. Other Methods - Bisection

  22. Bisection on x3 – x = 0

  23. Other Methods - Secant

  24. Secant – Real Case

  25. Secant – Complex Case

  26. Other Methods – Steepest Descent f(x, y)=0 and g(x, y)=0 f(x, y)2 + g(x, y)2

  27. Other Methods – Steepest Descent Re(z)=0 and Im(z)=0 Re(z)2 + Im(z)2

  28. Steepest Descent

  29. Steepest Descent

  30. Steepest Descent

  31. Steepest Descent

  32. The End! • Thanks! • www.unf.edu/~ddreibel

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