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Newton-Raphson help!

  1. Sep 16, 2009 #1
    1. The problem statement, all variables and given/known data

    [Divide and average Method] Square root of 2 was computed using the formula

    Xi+1 = 1/2 ( Xi + 2/Xi).------------ (1)

    Derive this method from the Newton-Raphson Formula

    2. Relevant equations



    3. The attempt at a solution

    Im boggled at how to derive this solution.

    The equation for Newton Raphson is

    F'(Xi) = (F(Xi) - 0) / Xi -(Xi+1) ------------- (2)

    which can be rearranged to

    Xi+1 = Xi - F(Xi) / F'(Xi)--------- (3)

    does this mean that i take the derivative of the equation (1)?

    (Xi+1)' =1/2(Xi+2/Xi)
    = ??
     
  2. jcsd
  3. Sep 17, 2009 #2

    CEL

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    You have
    [tex]x^2 = 2[/tex]
    or
    [tex]x = \frac{2}{x}[/tex]
    So,
    [tex]f(x) = \frac{2}{x}[/tex]
    Derive your equations from there.
     
    Last edited: Sep 17, 2009
  4. Sep 17, 2009 #3
    I still dont get it, do i take the derivative of 2/x?

    that would be f(x)'=-2(1/x^2)
     
  5. Sep 17, 2009 #4

    CEL

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    What is the Newton-Raphson method?
     
  6. Sep 17, 2009 #5
    Good job guy.

    /s
     
  7. Sep 17, 2009 #6
    I am having the same problem.

    Newton Raphson method: Xof(i+1) = xi - f(x)/f(x)'

    it is used to find roots by iteration
     
  8. Sep 18, 2009 #7

    CEL

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    Write your equation in the form y = f(x).
    Calculate f'(x).
    Choose a starting value for x0.
    If y - f(x0) < tolerance then end
    else
    Calculate x1 using Newton-Raphson formula.
    Iterate
     
  9. Sep 19, 2009 #8
    I think there is sufficient confusion amoung these posts to warrent another (hopefully non-confusing) post

    thedc: For Newton-Raphson, you are looking for the zero of a function (F), hence, you need to express the function (F) such that F(x) = 0.

    In your original post, you desire to find the answer to [itex]x[/itex] for [itex]x = \sqrt{2}[/itex]. Consider the more genral solution for [itex]x[/itex] with [itex]x = \sqrt{A}[/itex] for some positive [itex]A[/itex].

    Question: How can we express a function, [itex]F(x)[/itex], such that it results in [itex]F(x)=0[/itex] for this problem?

    Answer: Look at the [itex]x = \sqrt{A}[/itex]. This is really the same as finding [itex]x^2[/itex] such that [itex]x^2 = A[/itex]. Hence, one selection of [itex]F(x)[/itex] might be [itex]F(x) = x^2 - A=0[/itex].

    This is the "F" that is needed in the N-R method. The iterates for the solution of x are as follows:

    [tex]
    x_{\nu+1} = x_{\nu}-\frac{F(x_{\nu})}{F'(x_{\nu})}
    [/tex]

    Here, [itex]F'(x)[/itex] is shorthand to mean [itex]\frac{d\,}{dx}F(x)[/itex]. Also, in your case, the vale of A is A=2. You will need an initial estimate [itex]x_{0}[/itex] to start this procedure.



    The key to achieve the end goal of your exercise is to do some algebra on the resulting iterate expresion once you take the derivative of F and substitute it into the expression.

    Hope this helps.
     
    Last edited: Sep 19, 2009
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