A numerical study is performed to assess the impact of different limited-memor- y BFGS (L-BFGS) diagonal preconditioner update formulae and scaling strategies on the minimization performance. The formulae studied are those of Gilbert and Lemaréchal (1989) and a generalized version of the recently proposed quasi-Cauchy formula of Zhu et al. (1999). The scaling strategies are those of Gilbert and Lemaréchal (1989) and a new approach that is proposed. This numerical study uses a large number of test problems from the MODULOPT, MINPACK-2 and CUTE collections. Some rather stringent criteria are used for the line-search and the convergence, and the minimization is often performed up to the point where no more progress is achievable. It is found that the quasi-Cauchy formula overall performs poorly and suffers from a tendency to generate search directions numerically orthogonal to the gradient one. The good performance and robustness of the scaled direct BFGS diagonal-preconditioner update formula proposed by Gilbert and Lemaréchal (1989) is confirmed by the results of our experiments. The direct BFGS formula with the new scaling approach proposed is found to be much less robust. However, it significantly outperforms all the other update formulae in some cases.