Given a functional equation of form $f(f(x))=T(x)$ is there any good ways to solve it numerically? If not then at least approximate in some small region $x\in(-a;a)$.

E.g. with the equation $f(f(x))=x+x^2$ one could prove that the solution is between $l(x)=|x+0.5|+0.5$ and $h(x)=x+x^2$. However I'm unsure how to proceed further.

I've found a solution for equations like these on arxiv. However it seems to converge very badly (or I'm doing something wrong).