The Aitken's extrapolation can be written as $$X^n = X_{n-2} + \dfrac{(X_{n-1}- X_{n-2})^2}{(X_{n-1}- X_{n-2})-(X_n- X_{n-1})}$$
Verify it?
And $X^n$ can be viewed as being defined recursively by $$ Z_{n+1}= Z_n + \frac{\bigl(g(Z_n)- Z_n\bigr)^2}{\bigl(g(Z_n)- Z_n\bigr)-\bigl(g(g(Z_n))- g(Z_n)\bigr)} $$ Verify it for $|g'(X_n)|≠0$ or $1$, and show that $Z_n$ converges quadratically to $r$ even if $|g'(X_n)|> 1$ and the original iteration $X_{n+1}= g(X_n)$ diverges.
I am supposed to use the hint: $g(x) = (x-r)h(r) + r$, where $h(r)= g'(r)≠0$.
Any help is appreciated.