Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Published electronically: March 6, 2000
MathSciNet review: 1710648
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • 1. H Behnke, M Plum, C Wieners: Eigenvalue inclusions via domain decomposition. Preprint, 1999.
  • 2. E B Davies. Spectral Theory and Differential Operators. Cambridge Univ. Press, 1995. MR 96h:47056
  • 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. P Diaconis and D W Stroock: Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1 (1991) 36-61. MR 92h:60103
  • 5. F Goerisch: Ein Stufenverfahren zur Berechnung von Eigenwertschranken. In `Numerical Treatment of Eigenvalue Problems' vol. 4 ISNM 83 (Ed. J Albrecht et al) pp 104-114. Birkhauser-Verlag, Basel, 1987. MR 90j:65058
  • 6. T Kato: Perturbation Theory of Linear Operators. Springer-Verlag, Berlin, Heidelberg, New York, 1966. MR 34:3324
  • 7. R Lohner: Verified solution of eigenvalue problems in ordinary differential equations. Unpublished manuscript, 1990.
  • 8. M Plum: Eigenvalue inclusions for second order ordinary differential operators by a numerical homotopy method. Z. Angew. Math. Phys. 41 (1990) 205-226. MR 91d:65116
  • 9. M Plum: Bounds for eigenvalues of second order elliptic differential operators. Z. Angew. Math. Phys. 42 (1991) 848-863. MR 92h:35167
  • 10. M Plum: Guaranteed numerical bounds for eigenvalues. In `Spectral Theory and Computational Methods of Sturm-Liouville Problems', eds. D Hinton and P W Schaefer. Marcel Dekker, New York, Basel, 1997. MR 98g:47018
  • 11. S Zimmermann and U Mertins: Variational bounds to eigenvalues of self-adjoint eigenvalue problems with arbitrary spectrum. Z. Anal. Anwendungen. 14 (1995) 327-345. MR 96d:49046

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 34L15, 35P15, 49R05, 49R10, 65L15, 65L60, 65L70, 65N25

Retrieve articles in all journals with MSC (1991): 34L15, 35P15, 49R05, 49R10, 65L15, 65L60, 65L70, 65N25

Additional Information

E. B. Davies
Affiliation: Department of Mathematics, King’s College, Strand, London, WC2R 2LS, United Kingdom

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

American Mathematical Society