Abstract
Popular methods like finite differences and finite elements approximate thekth eigenvalue of a regular Sturm-Liouville problem by thekth eigenvalue of a matrix problem of dimensionn. The accuracy deteriorates rapidly ask increases, and there is no approximation at all fork>n. Much recent work has been devoted to improving inexpensively the accuracy of the approximation with respect to increasingk. It is shown here that for many methods, it is possible with negligible cost and minimal complication to obtain approximations for allk that are uniformly accurate ink.
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