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
R
of prime characteristic. The methods used employ Lyubeznik’s
F
F
-modules over
R
R
, particularly his
F
F
-finite
F
F
-modules, and also the modules of generalized fractions of Sharp and Zakeri [Mathematika 29 (1982), pp. 32–41]. It is shown that many modules of generalized fractions over
R
R
have natural structures as
F
F
-modules; these lead to
F
F
-module structures on certain local cohomology modules over
R
R
, which are exploited, in conjunction with
F
F
-module structures on injective
R
R
-modules that result from work of Huneke and Sharp [Trans. Amer. Math. Soc. 339 (1993), pp. 765–779], to compute Lyubeznik numbers. The resulting algorithm has been implemented in Macaulay2.