(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The following is a modification of Newton's method:

x_{n+1}= x_{n}- f(x_{n}) / g(x_{n}) where g(x_{n}) = (f(x_{n}+ f(x_{n})) - f(x_{n})) / f(x_{n})

2. Relevant equations

We are supposed to use the following method:

let En = x_{n}+ p where p = root → x_{n}= p + En

Moreover, f(x_{n}) = f(p + En) = f(p) + f'(p)En + O(En^{2}) = f'(p)En + O(En^{2}) since f(p) = 0 at root

3. The attempt at a solution

Change the equation to x_{n+1}- x_{n}= - f(x_{n}) / g(x_{n}) where g(x_{n}) = (f(x_{n}+ f(x_{n})) - f(x_{n})) / f(x_{n})

Then,LHS:E_{n+1}- E_{n}

RHS broken up into parts:

f(x_{n}) = f(p + E_{n}) = f'(p)E_{n}+ O(E_{n}^{2})

f(x_{n}+ f(x_{n})) = f(p + E_{n}+ f(p + En)) = f(p + E_{n}+ f'(p)E_{n}+ O(E_{n}^{2})) = f(p) + f'(p)(E_{n}+ E_{n}f'(p) + O(E_{n}^{2})) + O(E_{n}^{2}) = f'(p)(E_{n}+ E_{n}f'(p) + O(E_{n}^{2})) + O(E_{n}^{2})

So, f(x_{n}+ f(x_{n})) - f(x_{n}) = [ f'(p)(E_{n}+ E_{n}f'(p) + O(E_{n}^{2})) + O(E_{n}^{2}) ] - f'(p)E_{n}+ O(E_{n}^{2})

= [ f'(p)(E_{n}f'(p) + O(E_{n}^{2})) + O(E_{n}^{2}) ] - O(E_{n}^{2})

Also, f(x_{n})^{2}= (f'(p)En + O(En^{2}))^{2}

So, E_{n+1}- E_{n}= (f'(p)En + O(En^{2}))^{2}/ ( [ f'(p)(E_{n}f'(p) + O(E_{n}^{2})) + O(E_{n}^{2}) ] - O(E_{n}^{2}) )

So, from this somehow I'm supposed to get E_{n+1}= O(E_{n}^{2})

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# Proving Quadratic Convergence via Taylor Expansion

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