Numerical solution of non self-adjoint Sturm Liouville problems and related systems

This paper gives the analysis and numerics underlying a shooting method for approximating the eigenvalues of non-self-adjoint Sturm Liouville problems . We consider even order problems with (equally divided) separated boundary conditions. The method can nd the eigenvalues in a rectangle and in a left half-plane. It combines the argument principle with the compound matrix me thod (using the Magnus expansion). In some cases the computational cost of compound matrices can be reduced by transforming to a second order vector S turm Liouville problem. We study the asymptotics of the solutions of the OD E for large absolute values of the eigenvalue parameter in order to calcula te the eigenvalues in a left half-plane. The method is applied to the Orr-S ommerfeld equation and other examples.