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Correction of finite difference eigenvalues of periodic Sturm-Liouville problems

  • Alan L. Andrew (a1)
Abstract
Abstract

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.

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[2] R. S. Anderssen and F. R. de Hoog , “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions”, BIT 24 (1984) 401412.

[3] A. L. Andrew , “Eigenvectors of certain matrices”, Linear Algebra Appl. 7 (1973) 151162.

[5] A. L. Andrew , “Correction of finite element eigenvalues for problems with natural or periodic boundary conditions”, BIT 28 (1988) 254269.

[7] A. L. Andrew and J. W. Paine , “Correction of Numerov's eigenvalue estimates”, Numer. Math. 47 (1985) 289300.

[8] A. L. Andrew and J. W. Paine , “Correction of finite element estimates for Sturm-Liouville eigenvalues”, Numer. Math. 50 (1986) 205215.

[9] G. Doherty , M. J. Hamilton , P. C. Burton and E. I. von Nagy-Felsobuki , “A numerical variational method for calculating accurate vibrational energy separations of small molecules and their ions”, Austral. J. Phys. 39 (1986) 749760.

[11] J. Paine , “A numerical method for the inverse Sturm-Liouville problem”, SIAM J. Sci. Stat. Comput. 5 (1984) 149156.

[12] J. W. Paine , F. R. de Hoog and R. S. Anderssen , “On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems”, Computing 26 (1981) 123139.

[13] M. Porter and E. L. Reiss , “A numerical method for ocean-acoustic normal modes”, J. Acoust. Soc. Amer. 76 (1984) 244252.

[14] J. H. Wilkinson and C. Reinsch , Handbook for automatic computation, Vol. II, Linear algebra (Springer, New York, 1971).

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The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
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