# Accelerated newton method

1. Aug 22, 2011

### talolard

1. The problem statement, all variables and given/known data
given $$f \in C^2$$ such that $$f(a)=f'(a)=0 ^f''(a)\neq 0$$ prove that the modified newton method $$x_{n+1}=x_n-2 \frac{f(x_n){f'(x_n)}$$ 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 $$g^{(m+1)}\neq 0$$ 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 $$g(x)=x-2 \frac{f(x){f'(x)}$$ satisfies $$g(a)=0 ^ g'(a)\neq 0$$
But using le'hospitals rule to find g(a) i have $$g(a)=a-2\frac{f'(a)}{f''(a)}=a \neq 0$$
Whats wrong here?
Thanks
Tal

Last edited by a moderator: Aug 22, 2011
2. Aug 22, 2011

### Staff: Mentor

Fixed your LaTeX. I'm assuming that you were using the symbol ^ to mean "and."

3. Aug 22, 2011

### 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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