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Correction of finite difference eigenvalues of periodic Sturm-Liouville problems

Published online by Cambridge University Press:  17 February 2009

Alan L. Andrew
Alan L. Andrew
Affiliation:
Mathematics Department, La Trobe University, Bundoora, Victoria 3083, Australia.
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Abstract

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Computation of eigenvalues of regular Sturm-Liouville problems with periodic or semiperiodic boundary conditions is considered. A simple asymptotic correction technique of Paine, de Hoog and Anderssen is shown to reduce the error in the centred finite difference estimate of the kth eigenvalue obtained with uniform step length h from O(k4h2) to O(kh2). Possible extensions of the results are suggested and the relative advantages of the method are discussed.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 4 , April 1989 , pp. 460 - 469
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Anderssen, R. S. and de Hoog, F. R., “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions”, Tech. Rep. CMA-R05–82, Centre for Math. Analysis, Australian National Univ., 1982.Google Scholar
[2]Anderssen, R. S. and de Hoog, F. R., “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions”, BIT 24 (1984) 401–412.Google Scholar
[3]Andrew, A. L., “Eigenvectors of certain matrices”, Linear Algebra Appl. 7 (1973) 151–162.Google Scholar
[4]Andrew, A. L., ‘Asymptotic correction of finite difference eigenvalues”, in Computational techniques and applications: CTAC-85 (eds. Noye, J. and May, R..) (North-Holland, Amsterdasn, 1986) 333–341.Google Scholar
[5]Andrew, A. L., “Correction of finite element eigenvalues for problems with natural or periodic boundary conditions”, BIT 28 (1988) 254–269.CrossRefGoogle Scholar
[6]Andrew, A. L., “Some recent developments in finite element eigenvalue computation’, in Computational techniques and applications: CTAC-87 (eds. Noye, J. and Fletcher, C..) (North-Holland, Amsterdam, 1988) 83–91.Google Scholar
[7]Andrew, A. L. and Paine, J. W., “Correction of Numerov's eigenvalue estimates”, Numer. Math. 47 (1985) 289–300.CrossRefGoogle Scholar
[8]Andrew, A. L. and Paine, J. W., “Correction of finite element estimates for Sturm-Liouville eigenvalues”, Numer. Math. 50 (1986) 205–215.Google Scholar
[9]Doherty, G., Hamilton, M. J., Burton, P. C. and von Nagy-Felsobuki, E. I., “A numerical variational method for calculating accurate vibrational energy separations of small molecules and their ions”, Austral. J. Phys. 39 (1986) 749–760.Google Scholar
[10]Eastham, M. S. P., The spectral theory of periodic differential equations (Scottish Academic Press, Edinburgh, 1973).Google Scholar
[11]Paine, J., “A numerical method for the inverse Sturm-Liouville problem”, SIAM J. Sci. Stat. Comput. 5 (1984) 149–156.Google Scholar
[12]Paine, J. W., de Hoog, F. R. and Anderssen, R. S., “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems”, Computing 26 (1981) 123–139.Google Scholar
[13]Porter, M. and Reiss, E. L., “A numerical method for ocean-acoustic normal modes”, J. Acoust. Soc. Amer. 76 (1984) 244–252.Google Scholar
[14]Wilkinson, J. H. and Reinsch, C., Handbook for automatic computation, Vol. II, Linear algebra (Springer, New York, 1971).CrossRefGoogle Scholar