newton’s method finds zeros efficiently finds zeros of an equation: –solves f(x)=0 why do we...
TRANSCRIPT
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?)
2 2 4 6
60
40
20
20
40
60
Newton’s Method: Graphical Form
2 2 4 6
60
40
20
20
40
60
This is the function
It has only one zero, at
x = ??
2 2 4 6
60
40
20
20
40
60
This is the function
It has only one zero, at
x = 1
2 2 4 6
60
40
20
20
40
60
Newton’s Method is as follows:
1) Guess a point. Let’s use xo=4.
2 2 4 6
60
40
20
20
40
60
Newton’s Method is as follows:
1) Guess a point. Let’s use xo=4.
2) At that point on the graph,
2 2 4 6
60
40
20
20
40
60
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.
2 2 4 6
60
40
20
20
40
60
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:
2 2 4 6
60
40
20
20
40
60
Newton’s Method is as follows:
1) Guess a point. Let’s use xo=4.
2) At that point on the graph,
Draw the tangent.
3) Follow the tangent to the x-axis:
That’s our new guess.
2 2 4 6
60
40
20
20
40
60
Repeat those steps,
Until the answer doesn’t change:
That’s the root!
2 2 4 6
60
40
20
20
40
60
Remember our steps:
1) Guess a point: in this case, xo=4.
Newton’s Method: Algebraic Form
2 2 4 6
60
40
20
20
40
60
Remember our steps:
1) Guess a point: in this case, xo=4.
(xo,0)
2 2 4 6
60
40
20
20
40
60
Remember our steps:
1) Guess a point: in this case, xo=4.
2) At that point on the graph,
(xo,0)
(xo, f(xo))
2 2 4 6
60
40
20
20
40
60
Remember our steps:
1) Guess a point: in this case, xo=4.
2) At that point on the graph,
Draw the tangent.
(xo,0)
(xo, f(xo))Slope: f´(xo)
2 2 4 6
60
40
20
20
40
60
Remember our steps:
1) Guess a point: in this case, xo=4.
2) At that point on the graph,
Draw the tangent.
3) Follow the tangent to the x-axis:
That’s our new guess.
(xo,0)
(xo, f(xo))Slope: f´(xo)
(xo-??,0)
2 2 4 6
60
40
20
20
40
60
(xo,0)
(xo, f(xo))Slope: f´(xo)
(xo-??, f(xo)-f(xo))
2 2 4 6
60
40
20
20
40
60
(xo,0)
(xo, f(xo))Slope: f´(xo)
2 2 4 6
60
40
20
20
40
60
(xo,0)
(xo, f(xo))Slope: f´(xo)
Newton’s Method iterates to find a zero:
(“iterate” means feed the answer
back in to find the next answer)
At each step,
2 2 4 6
60
40
20
20
40
60
(xo,0)
(xo, f(xo))
Newton’s Method iterates to find a zero:At each step,
(x2,0)(x3,0) (x1,0)
4 2 2 4
20
10
10
20
30
40
50
Your Turn
f(x)=(x+3)(x+1)(x-1)(x-3)
Your Turn
4 2 2 4
20
10
10
20
30
40
50
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?
4 2 2 4
20
10
10
20
30
40
50
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 GuessesLet’s extend to complex plane: look at function
• Where does this have zeros?
4 2 2 4
20
10
10
20
30
40
50
f(x)=(x+3)(x+1)(x-1)(x-3)
-2 -1.5 -1 -0.5 0.5 1 1.5 2
-2
-1.5
-1
-0.5
0.5
1
1.5
2
Complex Map of GuessesLet’s extend to complex plane: look at function
• Where does this have zeros? (At 1, -1, i, -i)
Complex Map of GuessesYou might think the map of guesses looks like this:
Complex Map of GuessesIn 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:
http://xaos.sourceforge.net/