We apply generalized cross-validation (GCV) as a stopping rule for general linear stationary iterative methods for solving very large-scale, ill-conditioned problems. We present a new general formula for the influence operator for these methods and, using this formula and a Monte Carlo approach, we show how to compute the GCV function at a cheaper cost. Then we apply our approach to a well known iterative method (ART) with simulated data in positron emission tomography (PET).122417433