Given an equation such as $4x = e^x $, I remember from high school that we pick some initial point $x = x_0$ and iteratively evaluate $x_{n+1} = \frac{1}{4} e^{x_n} $ until convergence. If the solution does not converge, try different initial points.
What is the name of such class of methods (iterative method? ) and why do these work?