  Denote by $T$ the transformation $T(x)= 2 \,x $ (mod 1). Given a potential
$A:S^1 \to \mathbb{R}$ we are interested in exhibiting in several examples the
explicit expression for the calibrated subaction $V: S^1 \to \mathbb{R}$ for
$A$. The action of the $1/2$ iterative procedure $\mathcal{G}$, acting on
continuous functions $f: S^1 \to \mathbb{R}$, was analyzed in a companion
paper. Given an initial condition $f_0$, the sequence, $\mathcal{G}^n(f_0)$
will converge to a subaction. The sharp numerical evidence obtained from this
iteration allow us to guess explicit expressions for the subaction in several
worked examples: among them for $A(x) = \sin^2 ( 2 \pi x)$ and $A(x) = \sin ( 2
\pi x)$. Here, among other things, we present piecewise analytical expressions
for several calibrated subactions. The iterative procedure can also be applied
to the estimation of the joint spectral radius of matrices. We also analyze the
iteration of $\mathcal{G}$ when the subaction is not unique. Moreover, we
briefly present the version of the $1/2$ iterative procedure for the estimation
of the main eigenfunction of the Ruelle operator.
