We put forward a new method for the solution of eigenvalue problems for (systems of) ordinary differential equations, where our main focus is on eigenvalue problems for singular Schrödinger equations arising, for example, in electronic structure computations. Here, the generation of the starting values for the computation of eigenvalues of higher index is a critical issue. Our approach comprises two stages: First we generate rough approximations by a matrix method. These approximations are used

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... as starting values for an adaptive collocation method which yields approximations of high accuracy together with a reliable error estimate. We successfully apply our method to the solution of the coupled ODE Stark problem for the hydrogen atom.