  This paper presents an algorithm for calculation of the Lyubeznik numbers of
a local ring which is a homomorphic image of a regular local ring $R$ of prime
characteristic. The methods used employ Lyubeznik's $F$-modules over $R$,
particularly his $F$-finite $F$-modules, and also the modules of generalized
fractions of Sharp and Zakeri. It is shown that many modules of generalized
fractions over $R$ have natural structures as $F$-modules; these lead to
$F$-module structures on certain local cohomology modules over $R$, which are
exploited, in conjunction with $F$-module structures on injective $R$-modules
that result from work of Huneke and Sharp, to compute Lyubeznik numbers. The
resulting algorithm has been implemented in Macaulay2.
