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Homework Help: Finding the roots through Newtons method

  1. Aug 14, 2010 #1
    1. The problem statement, all variables and given/known data
    In each of the following items approximate the zeros of [tex]f[/tex] using Newton's method. Continue iterating until making two successive approximations differ at most in 0.001.

    The Newton iteration: [tex]x_{n+1}=x_n-\displaystyle\frac{f(x_n)}{f'(x_n)}[/tex]

    Well, I have a doubt about this. I'm not sure if it's asking me to iterate till [tex]|x_{n+1}-x_n|\leq{0.001}|[/tex], or if I should apply some of this:

    [tex]k_1>0[/tex], [tex]|f'(x)|\geq{k_1}[/tex] and [tex]|f''(x)|\leq{}k_2[/tex] for all [tex]x\in{}[/tex], then:

    [tex]|x_{n+1}-r|<\displaystyle\frac{k_2}{2k_1}|x_n-r|^2[/tex]

    If [tex]r\in{}[r-\delta,r+\delta]\subset{[a,b]}[/tex], and [tex]\delta<2(\displaystyle\frac{k_1}{k_2})[/tex]

    [tex]|x_{n+1}-r|<\displaystyle\frac{2k_1}{k_2}(\displaystyle\frac{\delta}{\displaystyle\frac{2k_1}{k_2}})^2n[/tex]
     
    Last edited: Aug 14, 2010
  2. jcsd
  3. Aug 14, 2010 #2
    Stop when [tex]|x_{n+1} - x_n| < 0.001[/tex].
     
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