This paper is concerned with the solution of the nonlinear system of equations arising from the A.M. Il'in's scheme approximation of a model semilinear singularly perturbed boundary value problem. We employ Newton and Picard methods and propose a new version of the two-grid method originated by O. Axelsson [2] and J. Xu [19]. In the first step, the nonlinear differential equation is solved on a "coarse" grid of size $H$. In the second step, the problem is linearized around an appropriate interpolation of the solution computed in the first step and the linear problem is then solved on a fine grid of size $h<<H$. It is shown that the algorithms achieve optimal accuracy as long as the mesh sizes satisfy $h = O(H^{2^m})$, $m=1,2,...$, where $m$ is the number of the Newton (Picard) iterations for the difference problem. We count the number of the arithmetical operations to illustrate the computational cost of the algorithms. Numerical experiments are discussed.