  Let $X$ be a smooth projective surface such that linear and numerical
equivalence of divisors on $X$ coincide and let $\sigma\subseteq |D|$ be a
linear pencil on $X$ with integral general fibers. A fiber of $\sigma$ will be
called special if either it is not integral or it has non-generic multiplicity
at some of the base points (including the infinitely near ones) of the pencil.
In this note we provide an algorithm to compute the integral components of the
special fibers of $\sigma$.
