Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important to select fast, robust and reliable methods for their solution, also in the case where fast modern computers are available. Since the coefficient matrices of the systems are normally sparse (i.e. most of their elements are zeros), the first requirement is to efficiently exploit the sparsity. However, this is normally not sufficient when the systems are very large. The computation of preconditioners based on approximate LU-factorizations and their use in the efforts to increase further the efficiency of the calculations will be discussed in this paper. Computational experiments based on comprehensive comparisons of many numerical results that are obtained by using ten well-known methods for solving systems of linear algebraic equations (the direct Gaussian elimination and nine iterative methods) will be reported. Most of the considered methods are preconditioned Krylov subspace algorithms.