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Asymptotic correction of Numerov’s eigenvalue estimates with natural boundary conditions. (English) Zbl 0970.65086

The method of asymptotic correction was first studied by this author to improve numerical approximations to eigenvalues of regular Sturm-Liouville problems

$-{y}^{\text{'}\text{'}}+qy=\lambda y,\phantom{\rule{1.em}{0ex}}y\left(a\right)=y\right)\pi \right)=0·$

The key idea of this method is that, at least for sufficiently $q$ and for certain classes of finite difference and finite element schemes, the leading asymptotic term in the error is a computed eigenvalue is independent of $a$. Asymptotic correction, at negligible extra cost, can greatly improve the accuracy of approximations to higher eigenvalues. Here, the method is extended to problems with natural boundary conditions to improve approximations obtained by Numerov’s finite difference scheme.

##### MSC:
 65L15 Eigenvalue problems for ODE (numerical methods) 65L12 Finite difference methods for ODE (numerical methods) 34L16 Numerical approximation of eigenvalues and of other parts of the spectrum 34B24 Sturm-Liouville theory