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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=λy,α 1 y ' (0)-α 2 y(0)=β 1 y ' (π)+β 2 y(π)=0, obtained by the standard centered difference method with step length h, is O(k 4 h 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 α 1 =β 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 α 1 and β 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, =O(k -1 ), Φ ˜=O(k -1 ), require the condition α 1 0. 2. The role of the additional parameter α 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:
65L15Eigenvalue problems for ODE (numerical methods)
34L99Ordinary differential operators
References:
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[9]J. W. Paine, F. R. de Hoog and R. S. Anderssen,On the correction of finite difference eigenvalue approximation for Sturm-Liouville problems, Computing 26 (1981), 123–139. · Zbl 0445.65087 · doi:10.1007/BF02241779
[10]B. N. Parlett,The Symmetric Eigenvalue Problem, Englewood Cliffs, Prentice-Hall, N.J. (1980).
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