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On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions. (English) Zbl 0552.65065

The error in the approximation to the kth eigenvalue of \(-y''+qy=\lambda y,\alpha_ 1y'(0)-\alpha_ 2y(0)=\beta_ 1y'(\pi)+\beta_ 2y(\pi)=0,\) obtained by the standard centered difference method with step length h, is \(O(k^ 4h^ 2)\). A major improvement was made by J. W. Paine and the authors [Computing 26, 123-139 (1981; Zbl 0436.65063)] who showed that, in the case \(\alpha_ 1=\beta_ 1=0\), a simple correction reduced the error to \(0(kh^ 2)\). The present paper makes two further significant advances: the correction technique is extended to general \(\alpha_ 1\) and \(\beta_ 1\), and it is proved that the error in the corrected eigenvalues is \(O(h^ 2)\), i.e. it is independent of k.
{Reviewer’s comments: 1. The asymptotic formulae, \(\emptyset =O(k^{- 1}),\) \({\tilde \Phi}=O(k^{-1}),\) require the condition \(\alpha_ 1\neq 0\). 2. The role of the additional parameter \(\alpha\) in the main theorem is clarified in a subsequent paper of the reviewer and J. W. Paine [Numer. Math. (to appear)] which examines a similar correction for Numerov’s method.}
Reviewer: A.L.Andrew

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34L99 Ordinary differential operators

Citations:

Zbl 0436.65063
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References:

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