IN the simulation on an electronic computer of the acceleration process in a high-current ion linear accelerator [1] the greater part of the computer time is spent on the numerical solution at each step of Poisson's equation in a cylindrical co-ordinate system. The domain ( −R ⩽r $ ̌ R,0 ⩽z⩽ L ), where the solution is sought, is usually such that Roliver, and hence in the numerical solution of the difference analogue of Poisson's equation the number of nodes in the direction Or is much less than the number of nodes in the direction 0 z. In this paper the method described in [2] is considered. In a Cartesian co-ordinate system with a corresponding choice of the iteration parameters, the determination with accuracy e of the solution of the system of difference equations by this method requires altogether O (In (1/ϵ) In ( 1 h )) iterations, where h is the step of the quadratic net. Moreover, it is known that after completing 1 h iterations it is possible to obtain the exact solution of the difference analogue problem. These results were obtained by the componentwise suppression of the error committed in the initial approximation. They are also easily generalised to the case of a cylindrical co-ordinate system, when the solution is sought in a rectangular domain and the eigenvalues of the difference operator approximating the second derivative with respect to z are used to suppress the error. To achieve economy of computer time in the cases where R⪡ L, it makes sense to choose the iteration parameters by the method described in [2] based on the properties of the difference operator approximating the differential operator ( 1 r) ( ∂/ ∂r) r( ∂ / ∂r). The present paper is devoted to questions connected with the choice of such parameters.