Finite difference approximations of the form L;:__,; d1,iui+i = L:' :! 1 e1.J(z1.i) for the numerical solution of linear nth order ordinary differential equations are analyzed. The order of these approximations is shown to be at least r 1 + s 1 + m 1 -n, and higher for certain special choices of the points ZJ,i· Similar approximations to initial or boundary conditions are also considered and the stability of the resulting schemes is investigated.