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Uniform estimation of the eigenvalues of Sturm–Liouville problems

Published online by Cambridge University Press:  17 February 2009

John Paine  and
Frank de Hoog
John Paine
Affiliation:
Computing Research Group, Australian National University, P.O. Box 4, Canberra, A.C.T. 2600, Australia
Frank de Hoog
Affiliation:
C.S.I.R.O. Division of Mathematics and Statistics, Yarralumla, A.C.T. 2600, Australia
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Abstract

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The perturbation of the eigenvalues of a regular Sturm–Liouville problem in normal form which results from a small perturbation of the coefficient function is known to be uniformly bounded. For numerical methods based on approximating the coefficients of the differential equation, this result is used to show that a better bound on the error is obtained when the problem is in normal form. A method having a uniform error bound is presented, and an extension of this method for general Sturm–Liouville problems is proposed and examined.

Type
Research Article
Information
The ANZIAM Journal , Volume 21 , Issue 3 , January 1980 , pp. 365 - 383
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Anderssen, R. S. and Cleary, J. R., “Asymptotic structure in torsional free oscillations of the earth—I. Overtone structure”, Geophys. J.R. Astr. Soc. 39 (1974), 241–268.CrossRefGoogle Scholar
[2]Birkhoff, O., de Boor, C., Swartz, B. and Wendroff, B., “Rayleigh-Ritz approximation by piecewise cubic polynomials”, SIAM J. Numer. Anal. 3 (1966), 188–203.CrossRefGoogle Scholar
[3]Canosa, J. and de Oliveira, R. G., “A new method for the solution of the Schrödinger equation”, J. Comp. Phys. 5 (1970), 188–207.CrossRefGoogle Scholar
[4]Coddington, E. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill, New York, 1955).Google Scholar
[5]Conte, S. D. and de Boor, C., Elementary numerical analysis: an algorithmic approach (McGraw-Hill, New York, 1972).Google Scholar
[6]Courant, R. and Hubert, D., Methods of mathematical physics, Vol. 1 (Interscience, New York, 1953).Google Scholar
[7]Gordon, R., “New method for constructing wave functions for bound states and scattering”, J. Chem. Phys. 51 (1969), 14–25.CrossRefGoogle Scholar
[8]Gregory, R. T. and Karney, D. T., A collection of matrices for testing computational algorithms (Wiley-Interscience, New York, 1969).Google Scholar
[9]Haskell, N. A., “The dispersion of surface waves on multilayered media”, Bull. Seisinol. Soc. Amer. 43 (1953), 17–34.CrossRefGoogle Scholar
[10]Ixaru, L. Gr., “The error analysis of the algebraic method for solving the Schrödinger equation”, J. Comp. Phys. 9 (1972), 159–163.CrossRefGoogle Scholar
[11]Kato, T., Perturbation theory for linear operators (Springer-Verlag, Berlin, 1966).Google Scholar
[12]Osborne, M. R., “Numerical procedures for the eigenvalue problems of second order ordinary differential equations when a wide range of eigenvalues are required”. In Actas del seminario sobre métodos numéricos modernos, Vol. 2 (Caracas, 1974).Google Scholar
[13]Osborne, M. R. and Michaelson, S., “The numerical solution of eigenvalue problems in which the eigenvalue parameter appears non-linearly, with an application to differential equations”, Computer J. 7 (1964), 66–71.CrossRefGoogle Scholar
[14]Pruess, S., “Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation”, SIAM J. Numer. Anal. 10 (1973), 55–68.CrossRefGoogle Scholar
[15]Pruess, S., “High order approximations to Sturm-Liouville eigenvalues”, Numer. Math. 24 (1975), 241–247.CrossRefGoogle Scholar
[16]Schultz, M. H., Spline analysis (Prentice Hall, Englewood Cliffs, N.J., 1973).Google Scholar
[17]Stone, M. H., Linear transformations in Hilbert space (American Mathematical Society, New York, 1932).Google Scholar
[18]Wang, C., Gettrust, J. F. and Cleary, J. R., “Asymptotic overtone structure in eigenfrequencies of torsional normal modes of the earth: a model study”, Geophys. J.R. Astr. Soc. 50 (1977), 289–302.CrossRefGoogle Scholar