Asymptotic expansions for the extended systems used in the calculation of some singularities in b.v.p's

The numerical computation of generalized turning points is often obtained by replacing the original system by an extended problem for which the singular point is an isolated solution. We will show that there exists an asymptotic expansion in even powers of h (the step size) for the discretization error of the approximate solution of the extended system. This will mean that it is possible to improve the accuracy of the results by applying an extrapolation scheme. For numerical experimentation, we will consider two point boundary value problems. Finite differences will be used to discretize them. Results with small h will be improved using Richardson extrapolation.