  We describe a procedure based on the iteration of an initial function by an
appropriated operator, acting on continuous functions, in order to get a fixed
point. This fixed point will be a calibrated subaction for the doubling map on
the circle and a fixed Lipschitz potential. We study analytical and generic
properties of this process and we provide some computational evaluations of
subactions using a discretization of the circle. The fixed point is unique if
the maximizing probability is unique. We proceed a careful analysis of the
dynamics of this operator close by the fixed point in order to explain the
difficulty in estimating its asymptotic behavior. We will show that the
convergence rate can be in some moments like $1/2$ and sometimes arbitrarily
close to $1$.
