Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators
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- by Alfred Carasso PDF
- Math. Comp. 23 (1969), 717-729 Request permission
Abstract:
In this paper we analyze the convergence of a centered finite-difference approximation to the nonselfadjoint Sturm-Liouville eigenvalue problem where $[{\text {unk}}]$ has smooth coefficients and $a(x) \geqq {a_0} > 0$ on [0, 1]. We show that the rate of convergence is $O(\Delta {x^2})$ as in the selfadjoint case for a scheme of the same accuracy. We also establish discrete analogs of the Sturm oscillation and comparison theorems. As a corollary we obtain the result \[ \lim \sup \limits _{M \to \infty ;{\Delta _x} \to 0;(M + 1){\Delta _x} = 1} \left \{ {\sum \limits _{p = 1}^M {\frac {{||{V^p}||\infty }} {{{\Lambda _p}}}} } \right \} < \infty \] ) where $\Delta x = 1/(M + 1)$ is the mesh size and ${\Lambda _p},{V^p}$ are the characteristic pairs of $L$, the $M \times M$ matrix which approximates $[{\text {unk}}]$, and ${V^p}$ is normalized so that $||{V^p}|{|_2} = 1$.References
- H. Bückner, Über Konvergenzsätze, die sich bei der Anwendung eines Differenzenverfahrens auf ein Sturm-Liouvillesches Eigenwertproblem ergeben, Math. Z. 51 (1948), 423–465 (German). MR 30820, DOI 10.1007/BF01185778 A. Carasso, An Analysis of Numerical Methods for Parabolic Problems Over Long Times, Ph.D. Thesis, Univ. of Wisconsin, Madison, Wis., 1968.
- George E. Forsythe and Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR 0130124 F. R. Gantmacher, Matrix Theory, Vol. II, Chelsea, New York, 1964.
- F. R. Gantmacher and M. G. Kreĭn, Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Mathematische Lehrbücher und Monographien, I. Abteilung, Bd. V, Akademie-Verlag, Berlin, 1960 (German). Wissenschaftliche Bearbeitung der deutschen Ausgabe: Alfred Stöhr. MR 0114338
- John Gary, Computing eigenvalues of ordinary differential equations by finite differences, Math. Comp. 19 (1965), 365–379. MR 179926, DOI 10.1090/S0025-5718-1965-0179926-X
- Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 0175290
- Herbert B. Keller, On the accuracy of finite difference approximations to the eigenvalues of differential and integral operators, Numer. Math. 7 (1965), 412–419. MR 189279, DOI 10.1007/BF01436255
- Seymour V. Parter, Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations, Trans. Amer. Math. Soc. 99 (1961), 153–192. MR 120492, DOI 10.1090/S0002-9947-1961-0120492-5
- Seymour V. Parter, On the extreme eigenvalues of Toeplitz matrices, Trans. Amer. Math. Soc. 100 (1961), 263–276. MR 138981, DOI 10.1090/S0002-9947-1961-0138981-6
- Seymour V. Parter, On the eigenvalues of certain generalizations of Toeplitz matrices, Arch. Rational Mech. Anal. 11 (1962), 244–257. MR 143039, DOI 10.1007/BF00253939
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
- E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757 R. Courant & D. Hilbert, Methoden der Mathematischen Physik, Vol. 1, Interscience, New York, 1953. MR 16, 426.
- Donald Greenspan, Introductory numerical analysis of elliptic boundary value problems, Harper & Row, Publishers, New York-London, 1965. MR 0179956
- Frank W. Sinden, An oscillation theorem for algebraic eigenvalue problems and its applications, Mitt. Inst. Angew. Math. Zürich 1954 (1954), no. 4, 57. MR 67073
- Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Math. Comp. 23 (1969), 717-729
- MSC: Primary 65.62
- DOI: https://doi.org/10.1090/S0025-5718-1969-0258291-7
- MathSciNet review: 0258291