An iterative method is proposed for the numerical solution of finite difference equations which arise from the linear two-point boundary value problem: \(-u''+q(x)u=f(x)\), \(u(a)=\alpha\), \(u(b)=\beta\). Here the solution of the linear finite difference equation \(Au=b\) is based on a suitable splitting of the matrix A and an alternating strategy similar to the one used in the ADI method of Peaceman-Rachford. It must be noticed that the proposed method can be applied to the solution of block tridiagonal systems derived from the discretization of elliptic boundary value problems. Finally, under suitable assumptions, the convergence of the method is proved and some numerical experiments are presented.