Theory and Computation of Discrete Sturm--Liouville Problems on Non-Uniform Grids via the Prüfer Transformation

We study a discrete version of the Sturm–Liouville eigenvalue problem on grids whose spacing may vary from point to point, using the discrete Prüfer transformation. We show that any eigenvalues of the problem are real and that there are finitely many of them. We then compare two numerical methods for computing the eigenvalues, regular shooting and Prüfer-based shooting, and find that the Prüfer method remains accurate on non-uniform grids where regular shooting loses accuracy or fails.