Author’s summary: The main subject of the paper is the study of the performance of successive overrelaxation (SOR) algorithms for solving linear systems of the type arising from the difference approximation of nonself-adjoint two-dimensional elliptic partial differential equations. A special attention is paid to the development of efficient techniques for determining the optimum relaxation parameter providing the maximum rate of convergence. Four models of the behaviour of the spectral radius of the SOR matrix as a function of relaxation parameter are analyzed. Numerical experiments are performed for several problems with nonsymmetric coefficient matrices taken from the literature. A comparison of results of the line-SOR method with the results obtained from different generalized minimal residual algorithms shows that with the computational work comparable for both methods, the line-SOR method provides the solutions of considered problems with the second norm of the error vector a few orders lesser in the magnitude.