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Accelerated newton method

  1. Aug 22, 2011 #1
    1. The problem statement, all variables and given/known data
    given [tex] f \in C^2 [/tex] such that [tex] f(a)=f'(a)=0 ^f''(a)\neq 0 [/tex] prove that the modified newton method [tex] x_{n+1}=x_n-2 \frac{f(x_n){f'(x_n)} [/tex] coverges with order two.


    2. Relevant equations
    if g(x) is an iterative function such that the first m derivatives of g at a are zero and [tex]g^{(m+1)}\neq 0 [/tex] then the order of convergence is m+2



    3. The attempt at a solution

    So it seems that i want to show that my iterating function [tex] g(x)=x-2 \frac{f(x){f'(x)} [/tex] satisfies [tex] g(a)=0 ^ g'(a)\neq 0 [/tex]
    But using le'hospitals rule to find g(a) i have [tex] g(a)=a-2\frac{f'(a)}{f''(a)}=a \neq 0 [/tex]
    Whats wrong here?
    Thanks
    Tal
     
    Last edited by a moderator: Aug 22, 2011
  2. jcsd
  3. Aug 22, 2011 #2

    Mark44

    Staff: Mentor

    Fixed your LaTeX. I'm assuming that you were using the symbol ^ to mean "and."
     
  4. Aug 22, 2011 #3

    Mark44

    Staff: Mentor

    Did you calculate g'(x)? You will need g'(x) so that you can evaluate g'(a). I'm not sure why you think you need L'Hopital's Rule.
     
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