An algebraic procedure for the spectral corrections using the miss-distance functions in regular and singular Sturm-Liouville problems

A general method based on the evaluation of the zeros of a suitable polynomial is suggested in order to have an estimation of the spectral error in the numerical treatment of Sturm-Liouville problems. The method is strictly concerned with the miss-distance function arising in the shooting algorithm for eigenvalues. The error correcting procedure derived from the method is particularly helpful when difficulties arise in the numerical integration. Two kinds of Sturm-Liouville problems are considered: the standard regular problems on a closed interval and the problems where an eigenvalue is nonlinearly involved and embedded in an essential spectrum giving origin to an inner singularity. Numerical experiments clearly highlight the efficaciousness of the proposed method both in the regular and singular case.