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)=x 3 -1=0: Guess a solution x 0 Is it right? Guess a second solution x 1 Is it right? Construct a third guess: x 2 =x 1 - (x 0 -x 1 )/(f(x 0 )-f(x 1 )) (This is where the secant through the first two points cuts the x axis) Repeat indefinitely.

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

7. x0x0 f(x 0 ) First guess: x 0

8. x0x0 f(x 0 ) Second guess: x 1 x1x1 f(x 1 )

9. Draw the secant and locate x 2 x2x2

10. Draw another secant and locate x 3 x2x2 x3x3 x1x1

12. Does this always work?

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

21. Complex Numbers What is the solution to x 2 = -1?

22. Complex Numbers -i 0 i -i 0 i -1 0 1 0 0 0 1 0 -1

23. Complex Numbers i  (0.5+i)

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

25. Complex Numbers 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.

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

28. Strategy 2: Choose x 0 freely, let x 1 be very close to x 0.

30. Newton’s Method To find the roots of f(x) = 0, construct the series {x i }, where x i+1 = x i – f(x i )/f / (x i ) (and x 0 is a random guess) Example: f(x) = x 3 -1, so f / (x) = 2x 2 x 0 = 2, so x 1 = 2 –(2 3 -1)/(2*2 2 ) = 2 – 7/8 = 1.125 and x 2 = 1.125-(1.125 3 -1)/(2*1.125 2 ) = 0.9575

31. Newton’s Method x0x0 f(x 0 )

32. Newton’s Method x0x0 x1x1

33. x0x0 x1x1 f(x 1 )

34. Newton’s Method x2x2 x1x1

35. Apply Newton’s method to z 3 -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.

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

42. We recognise the Mandelbrot set, which can also be generated by a simpler process: Repeat the calculation z n = z 2 n-1 +z 0 until z n > 2 or you give up. Colour in the complex point z=x+iy according to how long this took.

46. 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.

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