Insight is given into the conditions of derivative matrices to be inverted in point-relaxation methods for 1-D and 2-D, first-order upwind-discretized Euler equations. Speed regimes are found where ill-conditioning of these matrices occurs; 1-D flow equations appear to be less well conditioned than 2-D flow equations. The ill-conditioning is easily improved by adding regularizing matrices to the derivative matrices. A smoothing analysis is made of point Gauss-Seidel relaxation applied to discrete Euler equations conditioned by such an additive matrix. The method is successfully applied to a very low-subsonic, steady, 2-D stagnation flow.