Newtons Method

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Newtons Method

• Presented by Flip and Marcus

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Newtons Method finds Zeros

• Efficiently finds Zeros of an equation
• Solves f(x)0
• Why do we care?

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Newtons 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)-a0).
• Factor Polynomials
• Find minima and maxima (Where does f(x)0?)
• Find singular points (Where does 1/f(x) blow up?)

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Newtons Method Graphical Form
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This is the function It has only one zero, at
x ??
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This is the function It has only one zero, at
x 1
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Newtons Method is as follows 1) Guess a point.
Lets use xo4.
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Newtons Method is as follows 1) Guess a point.
Lets use xo4. 2) At that point on the graph,
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Newtons Method is as follows 1) Guess a point.
Lets use x4. 2) At that point on the
graph, Draw the tangent.
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Newtons Method is as follows 1) Guess a point.
Lets use x4. 2) At that point on the
graph, Draw the tangent. 3) Follow the tangent
to the x-axis
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Newtons Method is as follows 1) Guess a point.
Lets use xo4. 2) At that point on the
graph, Draw the tangent. 3) Follow the tangent
to the x-axis Thats our new guess.
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Repeat those steps, Until the answer doesnt
change Thats the root!
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Newtons Method Algebraic Form
Remember our steps 1) Guess a point in this
case, xo4.
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(xo,0)
Remember our steps 1) Guess a point in this
case, xo4.
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(xo, f(xo))
(xo,0)
Remember our steps 1) Guess a point in this
case, xo4. 2) At that point on the graph,
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(xo, f(xo))
Slope f(xo)
(xo,0)
Remember our steps 1) Guess a point in this
case, xo4. 2) At that point on the
graph, Draw the tangent.
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(xo, f(xo))
Slope f(xo)
(xo,0)
(xo-??,0)
Remember our steps 1) Guess a point in this
case, xo4. 2) At that point on the
graph, Draw the tangent. 3) Follow the tangent
to the x-axis Thats our new guess.
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(xo, f(xo))
Slope f(xo)
(xo,0)
(xo-??, f(xo)-f(xo))
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(xo, f(xo))
Slope f(xo)
(xo,0)
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(xo, f(xo))
Slope f(xo)
(xo,0)
Newtons Method iterates to find a
zero (iterate means feed the answer back
in to find the next answer) At each step,
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(xo, f(xo))
(x2,0)
(xo,0)
(x3,0)
(x1,0)
Newtons Method iterates to find a zeroAt each
step,
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Your Turn
Your Turn
f(x)(x3)(x1)(x-1)(x-3)
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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 Thats our new guess.
Repeat those steps, Until the result doesnt
change Thats the root!
f(x)(x3)(x1)(x-1)(x-3)
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Approximating Zeros

• Newtons Method isnt the only way
• Use 1 guess, derivative
• Newtons Method
• Use 2 guesses, interval must contain a zero
• Bisection Method
• Secant Method
• False Position Method
• Computers Calculators
• One of the interval methods

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Why does the TI-89 Lie?!

• Bust out your calculators and find
• TI-89 input x2 into the first y-input. Graph
  that equation. Then hit F5 followed by 2Zero and
  then type in -1 and 1. Wait 30 seconds or more.
• TI-83input x2 into the first y-input. Graph
  that equation. Push second Calc and then choose
  your bounds but do not choose 0 as your guess.

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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.

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How can we Break it?

• How can we make Newtons Method Fail?
• (Newtons Method
• Want to find roots of an equation
• Using an initial guess,
• Iterate the equation
• Until result doesnt change)

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How can we Break it?

• Ask a stupid question
• No real roots
• Roots at infinity
• Break the equation

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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 doesnt have derivative
• Function that doesnt change sign at root
• Equivalently derivative is zero at the root.
• Use a foolish initial guess
• What happens?

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What happens when you guess Foolishly?

• 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?

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Are there Foolish Guesses?

• Lets 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)(x3)(x1)(x-1)(x-3)
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Complex Map of Guesses

• Lets extend to complex plane look at function
• Where does this have zeros?

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f(x)(x3)(x1)(x-1)(x-3)
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Complex Map of Guesses

• Lets extend to complex plane look at function
• Where does this have zeros? (At 1, -1, i, -i)

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Complex Map of Guesses

• You might think the map of guesses looks like
  this

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Complex Map of Guesses

• In fact, it looks like this

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Complex Map of Guesses

• This is a Newtons 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/