Accurate computation of higher Sturm-Liouville eigenvalues
The computation of eigenvalues of regular Sturm-Liouville problems is considered. It is shown that a simple step-dependent linear multistep method can be used to reduce the error of the order \(k^ 4h^ 2\) of the centered finite difference estimate of the k-th eigenvalue with uniform step length h, to an error of order \(kh^ 2\). By an appropriate minimization of the local error term of the method one can obtain even more accurate results. A comparison of the simple correction techniques of \textit{J. W. Paine}, \textit{F. R. de Hoog}, and \textit{R. S. Anderssen} [Computing 26, 123--135 (1981; Zbl 0436.65063)] and of \textit{A. L. Andrew} and \textit{J. W. Paine} [Numer. Math. 50, 205--215 (1986; Zbl 0588.65062)] is given. Numerical examples demonstrate the usefulness of this correction even for low values of \(k\).
scientific article; zbMATH DE number 990