Illustration of the Newton–Raphson method starting from an initial seed (red dot) and reaching a solution of the lens equation (green dot) in five steps. The vector field indicates the direction of ϵ as calculated by Eq. (18). First, we note that the field locally converges to the solution. However, beyond a nearby critical curve (gray line in the bottom left) the vector flow diverges from this solution. This is a good reason to forbid critical curve crossing (Sect. 6.2.3).