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Chaos

Chaos. State-of-the-art calculator,1974 (about $400). State-of-the-art calculator, 2013 (about $40). How does the `solve ’ function work?. Research (looking in the manual) shows that it employs something called `the secant method ’. Using the secant method to solve f(x) = x 3 -1=0 :

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Chaos

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  1. Chaos

  2. State-of-the-art calculator,1974 (about $400)

  3. State-of-the-art calculator, 2013 (about $40)

  4. How does the `solve’ function work? Research (looking in the manual) shows that it employs something called `the secant method’.

  5. Using the secant method to solve f(x)=x3-1=0: Guess a solution x0 Is it right? Guess a second solution x1 Is it right? Construct a third guess: x2 =x1 - (x0-x1)/(f(x0)-f(x1)) (This is where the secant through the first two points cuts the x axis) Repeat indefinitely.

  6. f(x) x Find the point(s) at which f(x)=0

  7. f(x0) x0 First guess: x0

  8. f(x0) x1 x0 f(x1) Second guess: x1

  9. x2 Draw the secant and locate x2

  10. x1 x2 x3 Draw another secant and locate x3

  11. Does this always work?

  12. x1 Colour this point according to how long it takes to get to the right answer. x0 Showing the success of the secant method for many different pairs of initial guesses:

  13. Complex Numbers What is the solution to x2 = -1?

  14. Complex Numbers -i 0 i -1 0 1 0 0 0 1 0 -1 -i 0 i

  15. Complex Numbers i  (0.5+i)

  16. Complex Numbers Now the equation x3 - 1 = 0 has 3 roots: x=1, x=0.5+√3i/2, x=0.5-√3i/2

  17. The secant method doesn’t take us to the complex roots unless our initial guesses are complex. But now our initial two guesses have four components. Complex Numbers

  18. We flatten the tesseract by one of several strategies: 1. Let x0 be 0, choose x1 freely. Complex Numbers

  19. Strategy 2: Choose x0 freely, let x1 be very close to x0.

  20. Newton’s Method To find the roots of f(x) = 0, construct the series {xi}, where xi+1 = xi – f(xi)/f/(xi) (and x0 is a random guess) Example: f(x) = x3 -1, so f/(x) = 2x2 x0 = 2, so x1 = 2 –(23-1)/(2*22) = 2 – 7/8 = 1.125 and x2 = 1.125-(1.1253-1)/(2*1.1252) = 0.9575

  21. Newton’s Method f(x0) x0

  22. Newton’s Method x0 x1

  23. Newton’s Method x0 x1 f(x1)

  24. Newton’s Method x2 x1

  25. Apply Newton’s method to z3-1=0 which in the complex plane has three roots. Let the x and y axes represent the real and imaginary components of the initial guess. Colour them according to which root they reach, and when.

  26. One more equation to solve by Newton’s method: (x+1)(x-1)(x+ß)=0 …where ß is our first guess.

  27. We recognise the Mandelbrot set, which can also be generated by a simpler process: Repeat the calculation zn = z2n-1+z0 until zn > 2 or you give up. Colour in the complex point z=x+iy according to how long this took.

  28. Characteristics of Chaos Two ingredients-- non-linearity and feedback -- can give rise to chaos. Chaos is governed by deterministic rules, yet produces results that can be very hard to predict. Images of chaotic processes can display a high level of order, characterised by self-similarity.

  29. When can chaos arise? Trying to get two non-linear programs to converge: x y

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