It is obvious that for any given $f(x)$ there exists $g(x)$ such that $f(x)=0 \Leftrightarrow g(x)=x$. We could use this fact to solve any root finding problem using [fixed point iteration](http://en.wikipedia.org/wiki/Fixed_point_method) method, only if $g(x)$ promises convergence of the fixed point iteration method.


So I was wondering if there exists an algorithm that takes $f(x)$ and returns appropriate $g(x)$ and $x\_0$ such that $x\_{i+1}=g(x\_i)$ converges to a root of $f(x)$.



Convergence of fixed point iteration algorithm
You do not specify your class of $f$, and most importantly, do not specify where the convergence should hold, for which $x\_0$.


In general, the answer on your question is
Newton's method: $g(x)=x-f(x)/f'(x)$. The iterates usually converge when $x\_0$ is sufficiently close to the root, and if $f'(x)\neq 0$ at the root $x$.
Finding an algorithm with global convergence is another matter, and the answer strongly
depends on the class of functions $f$ you consider. There are even results on
non-existence of globally convergent iterative methods satisfying some natural conditions
for the class of rational functions (C. McMullen). 


