Bounds and higher-order estimates for Sturm-Liouville eigenvalues

Abstract

In [2] an algorithm is described for computing upper and lower bounds for the eigenvalues of the regular Sturm-Liouville problem on a finite interval [a, b]. The method, which deals easily with coefficients having jump discontinuities, replaces the coefftcients by constants on each subinterval of a partition rc := {a = x0 < x, < ... < x, = b). (See [I] for a survey of the related literature.) The bounds converge to the true eigenvalues as h := maxi hi -+ 0, where hi := xi xi1, the rate of convergence being of the first order in h[ 1, 21. However, it was noted in [2] that numerical results suggested that the arithmetic mean of the upper and lower bounds exhibits second-order convergence. In Section 3 we give a proof of this second-order convergence and also explain a remarkable numerical result noted in [2]. Although the main purpose of the algorithm of [2] is to give bounds, the fact that it simultaneously gives higher-order estimates is an important bonus.