Numerical experiments with applying approximate LU-factorizations as preconditioners for solving SLAEs with coefficient matrices from the "Sparse Matrix Market"

The solution of systems of linear algebraic equations (SLAEs) is very often the most time-consuming part of the computational process during the treatment of the original problems, 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 the solution of SLAEs when large applications are to be run, 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 exploit efficiently 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 we...