2. Newton’s Method finds Zeros Efficiently finds Zeros of an equation: –Solves f(x)=0 Why do we care?

3. Newton’s Method finds Zeros Efficiently finds Zeros of an equation: –Solves f(x)=0 Why do we care? –Can make any “solve for value” problem ( f(x)=a ) into a “find a zero” problem ( f(x)-a=0 ). –Factor Polynomials –Find minima and maxima (Where does f ´ (x)=0 ?) –Find singular points (Where does 1/ f(x) blow up?)

4. Newton’s Method: Graphical Form

5. This is the function It has only one zero, at x = ??

6. This is the function It has only one zero, at x = 1

7. Newton’s Method is as follows: 1) Guess a point. Let’s use x o =4.

8. Newton’s Method is as follows: 1) Guess a point. Let’s use x o =4. 2) At that point on the graph,

9. Newton’s Method is as follows: 1) Guess a point. Let’s use x=4. 2) At that point on the graph, Draw the tangent.

10. Newton’s Method is as follows: 1) Guess a point. Let’s use x=4. 2) At that point on the graph, Draw the tangent. 3) Follow the tangent to the x -axis:

11. Newton’s Method is as follows: 1) Guess a point. Let’s use x o =4. 2) At that point on the graph, Draw the tangent. 3) Follow the tangent to the x -axis: That’s our new guess.

12. Repeat those steps, Until the answer doesn’t change: That’s the root!

13. Remember our steps: 1) Guess a point: in this case, x o =4. Newton’s Method: Algebraic Form

14. Remember our steps: 1) Guess a point: in this case, x o =4. (x o,0)

15. Remember our steps: 1) Guess a point: in this case, x o =4. 2) At that point on the graph, (x o,0) (x o, f ( x o ) )

16. Remember our steps: 1) Guess a point: in this case, x o =4. 2) At that point on the graph, Draw the tangent. (x o,0) (x o, f ( x o ) ) Slope: f ´ ( x o )

17. Remember our steps: 1) Guess a point: in this case, x o =4. 2) At that point on the graph, Draw the tangent. 3) Follow the tangent to the x -axis: That’s our new guess. (x o,0) (x o, f ( x o ) ) Slope: f ´ ( x o ) (x o -??,0)

18. (x o,0) (x o, f ( x o ) ) Slope: f ´ ( x o ) (x o -??, f ( x o ) -f ( x o ) )

19. (x o,0) (x o, f ( x o ) ) Slope: f ´ ( x o )

20. (x o,0) (x o, f ( x o ) ) Slope: f ´ ( x o ) Newton’s Method iterates to find a zero: (“iterate” means feed the answer back in to find the next answer) At each step,

21. (x o,0) (x o, f ( x o ) ) Newton’s Method iterates to find a zero:At each step, (x 2,0) (x 3,0)(x 1,0)

22. Your Turn f(x)=(x+3)(x+1)(x-1)(x-3) Your Turn

23. 1) Start with the point written on your worksheet 2) At that point on the graph, Draw the tangent. 3) Follow the tangent to the x -axis: That’s our new guess. Your Turn Repeat those steps, Until the result doesn’t change: That’s the root! f(x)=(x+3)(x+1)(x-1)(x-3)

24. Approximating Zeros Newton’s Method isn’t the only way: –Use 1 guess, derivative Newton’s Method –Use 2 guesses, interval must contain a zero Bisection Method Secant Method False Position Method Computers & Calculators: –One of the interval methods

25. Why does the TI-89 Lie?! Bust out your calculators and find TI-89: input x^2 into the first y-input. Graph that equation. Then hit F5 followed by 2:Zero and then type in -1 and 1. Wait 30 seconds or more. TI-83:input x^2 into the first y-input. Graph that equation. Push second Calc and then choose your bounds but do not choose 0 as your guess.

26. Roots can be Dangerous! TI-83 uses numerical method combined with secants. TI-89 uses a complex algorithm that forms a rounding error from going from 14 decimal places to 16 decimal places back and forth.

27. How can we Break it? How can we make Newton’s Method Fail? (Newton’s Method: –Want to find roots of an equation –Using an initial guess, –Iterate the equation –Until result doesn’t change)

28. How can we Break it? Ask a stupid question –No real roots –Roots at infinity Break the equation:

29. How can we Break it? Ask a stupid question –No real roots –Roots at infinity Break the equation: –Function that is not continuous –Function that doesn’t have derivative –Function that doesn’t change sign at root Equivalently: derivative is zero at the root. Use a foolish initial guess –What happens?

30. Do all initial guesses go to a root? –Do some go off to infinity? –Do some bounce around forever? What root does each initial guess lead to? What happens when you guess Foolishly?

31. Are there Foolish Guesses? Let’s make a map: Each person grab a post-it note that corresponds to the root you found: Put it up, on the axis, at the point of your initial guess. f(x)=(x+3)(x+1)(x-1)(x-3)

32. Complex Map of Guesses Let’s extend to complex plane: look at function Where does this have zeros?

33. f(x)=(x+3)(x+1)(x-1)(x-3)

34. Complex Map of Guesses Let’s extend to complex plane: look at function Where does this have zeros? (At 1, -1, i, -i )

35. Complex Map of Guesses You might think the map of guesses looks like this:

36. Complex Map of Guesses In fact, it looks like this:

37. Complex Map of Guesses This is a “Newton’s Method” fractal Type of ‘Julia Set’ Fractal At each point of boundary, EVERY color touches! Program to explore this and other fractals: http://xaos.sourceforge.net/