A Note on the Relationship Between Finite-Difference and Shooting Methods for ODE Eigenvalue Problems

Finite-difference methods and shooting methods are two standard classes of techniques for solving ordinary differential equation eigenvalue problems. We first show that a Sturm sequence method often used for solving the finite-difference eigenvalue problem may be interpreted in the shooting framework. Then we find that the Sturm sequence procedure is easily extended to more complicated problems. This enables the shooting method to guarantee convergence to specified subsets of the modes of the problem without the usual requirement of an initial guess for the eigenvalues.