Fixpoint iteration of $ x + \ln(x) = 0 $

Question

I want to solve the equation $ x + \ln(x) = 0 $ which has a root $ x^* \approx 0.5$.

I am given three iterations:

$$ \quad x_{n+1} = - \ln(x_n)\\ \quad x_{n+1} = e^{-x_n} \\ x_{n+1} =\frac{x_n + e^{-x_n}}{2} $$

What method do I use to find equation 3?

Answer 1

I think it is a typo and should be $$x_{n+1} = \frac{x_n +e^{-x_n}}2$$

In this case the third iteration is a smoothed version of the second one.

We remember $1/2$ of the last estimate and add $1/2$ of a new estimate.