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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 $-{y}^{\text{'}\text{'}}+qy=\lambda y,{\alpha }_{1}{y}^{\text{'}}\left(0\right)-{\alpha }_{2}y\left(0\right)={\beta }_{1}{y}^{\text{'}}\left(\pi \right)+{\beta }_{2}y\left(\pi \right)=0,$ obtained by the standard centered difference method with step length h, is $O\left({k}^{4}{h}^{2}\right)$. 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 $0\left(k{h}^{2}\right)$. 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 $O\left({h}^{2}\right)$, i.e. it is independent of k.

{Reviewer’s comments: 1. The asymptotic formulae, $\varnothing =O\left({k}^{-1}\right),$ $\stackrel{˜}{{\Phi }}=O\left({k}^{-1}\right),$ require the condition ${\alpha }_{1}\ne 0$. 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 Eigenvalue problems for ODE (numerical methods) 34L99 Ordinary differential operators
##### References:
 [1] R. S. Anderssen and F. R. de Hoog,On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions, Centre for Mathematical Analysis Report CMA-R05-82, Australian National University (1982). [2] A. L. Andrew,Computation of higher Sturm-Liouville eigenvalues, Congressus Numerantium 34 (1982), 3–16. [3] F. R. de Hoog and R. S. Anderssen,Asymptotic formulas for the eigenvalues and eigenfunctions of continuous and discrete eigenvalue problems in Liouville normal form (in preparation). [4] G. Fix,Asymptotic eigenvalues of Sturm-Liouville systems, J. Math. Anal. Appl. 19 (1967), 519–525. · Zbl 0153.40401 · doi:10.1016/0022-247X(67)90009-1 [5] J. W. Paine,Numerical Approximation of Sturm-Liouville Eigenvalues, Ph.D. thesis, Australian National University (1979). [6] J. W. Paine and R. S. Anderssen,Uniformly valid approximation of eigenvalues of Sturm-Liouville problems in geophysics, Geophys. J. Roy. Astronom. Soc. 63 (1980), 441–465. [7] J. W. Paine and A. L. Andrew,Bounds on higher-order estimates for Sturm-Liouville eigenvalues, J. Math. Anal. Appl. 96 (1983), 388–394. · Zbl 0536.65066 · doi:10.1016/0022-247X(83)90048-3 [8] J. Paine and F. de Hoog,Uniform estimation of the eigenvalues of Sturm-Liouville problems, J. Austral. Math. Soc. (Ser. B) 21 (1980) 365–383. · Zbl 0417.34046 · doi:10.1017/S0334270000002459 [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). [11] S. Pruess,Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation, SIAM J. Numer. Anal. 10 (1973), 55–68. · Zbl 0259.65078 · doi:10.1137/0710008 [12] S. Pruess,Higher order approximations to Sturm-Liouville eigenvalues, Numer. Math. 24 (1975), 241–247. · Zbl 0305.65047 · doi:10.1007/BF01436595