A hierarchical method for obtaining eigenvalue enclosures
Author:
E. B. Davies
Journal:
Math. Comp. 69 (2000), 1435-1455
MSC (1991):
Primary 34L15, 35P15, 49R05, 49R10, 65L15, 65L60, 65L70, 65N25
DOI:
https://doi.org/10.1090/S0025-5718-00-01238-2
Published electronically:
March 6, 2000
MathSciNet review:
1710648
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
We introduce a new method of obtaining guaranteed enclosures of the eigenvalues of a variety of self-adjoint differential and difference operators with discrete spectrum. The method is based upon subdividing the region into a number of simpler regions for which eigenvalue enclosures are already available.
- 1. H Behnke, M Plum, C Wieners: Eigenvalue inclusions via domain decomposition. Preprint, 1999.
- 2. E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825
- 3. E B Davies. Spectral enclosures and complex resonances for general self-adjoint operators. LMS J. Comput. Math. 1 (1998) 42-74. CMP 98:15
- 4. Persi Diaconis and Daniel Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab. 1 (1991), no. 1, 36–61. MR 1097463
- 5. Friedrich Goerisch, Ein Stufenverfahren zur Berechnung von Eigenwertschranken, Numerical treatment of eigenvalue problems, Vol. 4 (Oberwolfach, 1986) Internat. Schriftenreihe Numer. Math., vol. 83, Birkhäuser, Basel, 1987, pp. 104–114 (German). MR 1023876
- 6. Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- 7. R Lohner: Verified solution of eigenvalue problems in ordinary differential equations. Unpublished manuscript, 1990.
- 8. Michael Plum, Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method, Z. Angew. Math. Phys. 41 (1990), no. 2, 205–226. MR 1045812, https://doi.org/10.1007/BF00945108
- 9. Michael Plum, Bounds for eigenvalues of second-order elliptic differential operators, Z. Angew. Math. Phys. 42 (1991), no. 6, 848–863. MR 1140697, https://doi.org/10.1007/BF00944567
- 10. Michael Plum, Guaranteed numerical bounds for eigenvalues, Spectral theory and computational methods of Sturm-Liouville problems (Knoxville, TN, 1996) Lecture Notes in Pure and Appl. Math., vol. 191, Dekker, New York, 1997, pp. 313–332. MR 1460558
- 11. S. Zimmermann and U. Mertins, Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum, Z. Anal. Anwendungen 14 (1995), no. 2, 327–345. MR 1337263, https://doi.org/10.4171/ZAA/677
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Additional Information
E. B. Davies
Affiliation:
Department of Mathematics, King’s College, Strand, London, WC2R 2LS, United Kingdom
Email:
E.Brian.Davies@kcl.ac.uk
DOI:
https://doi.org/10.1090/S0025-5718-00-01238-2
Keywords:
Spectrum,
eigenvalues,
spectral enclosures,
interval arithmetic,
Rayleigh-Ritz method,
Temple-Lehmann method
Received by editor(s):
October 27, 1998
Published electronically:
March 6, 2000
Article copyright:
© Copyright 2000
American Mathematical Society