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

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?)

Newton’s Method: Graphical Form

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

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

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

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

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.

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:

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.

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

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

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

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 ) )

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 )

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)

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

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

(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,

(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)

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

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)

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

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.

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.

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)

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

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?

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?

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)

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

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

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

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

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

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: