Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T02:37:06.733Z Has data issue: false hasContentIssue false
  • English
  • Français

Accelerated spectral refinement Part II: Cluster of eigenvalues

Published online by Cambridge University Press:  17 February 2009

Rafikul Alam ,
Rekha P. Kulkarni  and
Balmohan V. Limaye
Rafikul Alam
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay 400 076, India.
Rekha P. Kulkarni
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay 400 076, India.
Balmohan V. Limaye
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay 400 076, India.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The framework for accelerated spectral refinement for a simple eigenvalue developed in Part I of this paper is employed to treat the general case of a cluster of eigenvalues whose total algebraic multiplicity is finite. Numerical examples concerning the largest and the second largest multiple eigenvalues of an integral operator are given.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 2 , October 2000 , pp. 224 - 243
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Ahues, M., “A superlinearly convergent two-grid defect correction method for invariant subspaces of compact operators”, Numer. Funct. Anal. Optimiz. 8 (1985–1986) 599–621.Google Scholar
[2]Ahues, M., Aranchiba, S. and Telias, M., “Rayleigh-Schrödinger series for defective spectral elements of compact operators in Banach spaces. First part: Theoretical aspects”, Numer. Funct. Anal. Optimiz. 11 (1990–1991) 839–850.CrossRefGoogle Scholar
[3]Ahues, M. and Chatelin, F., “The use of defect correction to refine the eigenelements of compact integral operators”, SIAM J. Numer. Anal. 20 (1983) 1087–1093.CrossRefGoogle Scholar
[4]Ahues, M. and Largillier, A., “Rayleigh-Schrödinger series for defective spectral elements of compact operators in Banach spaces. Second part: Numerical comparison with some inexact Newton methods”, Numer. Funct. Anal. Optimiz. 11 (1990–1991) 851–872.Google Scholar
[5]Ahues, M. and Largillier, A., “Rayleigh-Schrödinger series versus inexact Newton methods for spectral computations”, in Iterative methods in linear algebra (eds. Beauwens, R. and de Groen, P.), (Elsevier Science Publishers B.V. North Holland, 1992) 415–422.Google Scholar
[6]Ahues, M. and Largillier, A., “A variant of fixed tangent method for spectral computations on integral operators”, Numer. Funct. Anal. Optimiz. 16 (1995) 1–17.CrossRefGoogle Scholar
[7]Ahues, M. and Telias, M., “Refinement methods of Newton type for approximate eigenelements of integral operators”, SIAM J. Numer. Anal. 23 (1986) 144–159.CrossRefGoogle Scholar
[8]Alam, R., Kulkarni, R. P and Limaye, B. V., “Boundedness of adjoint bases of approximate spectral subspaces and of associated block reduced resolvents”, Numer. Funct. Anal. Optimiz. 17 (1996) 473–502.Google Scholar
[9]Alam, R., Kulkarni, R. P and Limaye, B. V., “Accelerated spectral approximation”, Math. Comp. 67 (1998) 1401–1422.Google Scholar
[10]Alam, R., Kulkarni, R. P and Limaye, B. V., “Accelerated spectral refinement. Part I: Simple eigenvalue”, J. Austral. Math. Soc. (Series B) 41 (2000) 487–507.Google Scholar
[11]Dellwo, D., “Accelerated spectral refinement with application to integral operators”, SIAM J. Numer. Anal. 26 (1989) 1184–1193.CrossRefGoogle Scholar
[12]Limaye, B.V., “Spectral perturbation and approximation with numerical experiments”, in Proceeding of the Centre for Mathematical Analysis, Vol. 13, (Australian National University, 1986).Google Scholar
[13]Nair, M. T., “Computable error estimates for Newton's iterations for refining invariant subspaces”, Indian J. Pure Appl. Math. 21 (1990) 1049–1054.Google Scholar