Asymptotic correction and inverse eigenvalue problems: an overview

Abstract

Asymptotic correction was first used by Paine, de Hoog and Anderssen to improve the accuracy of finite difference approximations of higher Sturm--Liouville eigenvalues. Later it was used to develop an important class of methods for numerical solution of inverse Sturm--Liouville problems. It also shows promise as a method for the solution of more general inverse eigenvalue problems, including some involving partial differential equations and higher order operators. We critically review the literature on this subject and discuss some important open questions.