  The Giardin\`a-Kurchan-Peliti algorithm is a numerical procedure that uses
population dynamics in order to calculate large deviation functions associated
to the distribution of time-averaged observables. To study the numerical errors
of this algorithm, we explicitly devise a stochastic birth-death process that
describes the time evolution of the population probability. From this
formulation, we derive that systematic errors of the algorithm decrease
proportionally to the inverse of the population size. Based on this
observation, we propose a simple interpolation technique for the better
estimation of large deviation functions. The approach we present is detailed
explicitly in a two-state model.
