How quickly does Newton's Method for the approximation of the Lambert W function converge?
$$w_{j+1}=w_j-\frac{w_je^{w_j}-x}{e^{w_j}+w_je^{w_j}}$$
or simplified:
$$w_{j+1}=\frac{xe^{-w_j}+w_j^2}{w_j+1}$$
where $x$ replaces $z$ in the equation, because I'm only concerned with real numbers.
I understand that given $\infty$ iterations, it will converge to be absolutely accurate.
For example, after $n$ iteration, it will be accurate to $m$ decimal places.