# How should c be chosen to ensure rapid convergence of $x_{ n+1}= x_ n+c(f( x_ n))$ to $\alpha$?

Consider the rootfinding problem $f(x)=0$ with root $α$, with $f´(x)≠0$.

Convert it to the fixed-point problem $x=x+cf(x)≡g(x)$ with $c$ a nonzero constant.

How should c be chosen to ensure rapid convergence of $x_{n+1}=x_{n}+cf(x_{n})$ to α (Provided that $x_{0}$ is chosen sufficiently close to $α$)? Apply your way of choosing c to the rootfinding problem $x^3-5=0$

Does anyone could help me with this exercise please?

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Hint: The fixed point iteration for $g(x)=x$ behaves nicely if $|g'(x)|$ is small near the fixed point. Smallest possible is $0$. –  André Nicolas Nov 20 '12 at 20:53