The one-dimensional transport equation in slab geometry with periodic boundary conditions is studied. It reduces to the integral equation of the Peierls type. The spectral radius of the integral operator is estimated. We analyze the discrete-ordinates algorithm for estimating the solution. Convergence is proved and methods of estimating the rate of convergence are described. The estimates of some quadrature rules are derived by way of an example. The numerical results confirm the convergence properties of the proposed algorithm.