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Accelerated spectral refinement Part I: simple eigenvalue

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.
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Abstract

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A general framework is developed for constructing higher order spectral refinement schemes for a simple eigenvalue. Well-known techniques for ordinary spectral refinement are carried over to higher order spectral refinement yielding faster rates of convergence. Numerical examples are given by considering an integral operator.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 4 , April 2000 , pp. 487 - 507
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]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
[2]Ahues, M., d'Almeida, F., Chatelin, F. and Telias, M., “Iterative refinement techniques for the eigenvalue problem of compact integral operators”, in Treatment of Integral Equations by Numerical Methods (eds. Baker, C. T. H. and Miller, G. F.), (Academic Press, London, 1983) 373–386.Google Scholar
[3]Ahues, M., d'Almeida, F. and Telias, M., “Iterative refinement for approximate eigenelements of compact operators, R.A.I.R.O.”, Numer. Anal. 18 (1984) 125–135.Google Scholar
[4]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
[5]Alam, R., Kulkarni, R.P. and Limaye, B.V., “Boundedness of adjoint bases of approximate spectral subspaces and of associated block reduced resolvents”, Numer. Fund. Anal. Optimiz. 17 (1996) 473–502.CrossRefGoogle Scholar
[6]Alam, R., Kulkarni, R.P. and Limaye, B.V.. “Accelerated spectral approximation”, Mathematics of Computation 67 (1998) 1401–1422.CrossRefGoogle Scholar
[7]Chatelin, F., “Numerical computation of the eigenelements of linear integral operators by iterations”, SIAM J. Numer. Anal. 15 (1978) 1112–1124.CrossRefGoogle Scholar
[8]Chatelin, F., Spectral Approximation of Linear Operators (Academic Press, New York, 1983).Google Scholar
[9]Dellwo, D., “Accelerated spectral refinement with application to integral operators”, SIAM J. Numer. Anal. 26 (1989) 1184–1193.CrossRefGoogle Scholar
[10]Dellwo, D. and Friedman, M.B., “Accelerated spectral analysis of compact operators”, SIAM J. Numer. Anal. 21 (1984) 1115–1131.CrossRefGoogle Scholar
[11]Deshpande, L.N. and Limaye, B.V., “A fixed point technique to refine a simple approximate eigenvalue and a corresponding eigenvector”, Numer. Funct. Anal. Optimiz. 10 (1989) 909–921.CrossRefGoogle Scholar
[12]Gohberg, I., Goldberg, S. and Kaashoek, M.A., Classes of Linear Operators, Volume 1 (Birkhäuser-Verlag, Berlin, 1990).CrossRefGoogle Scholar
[13]Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials (Academic Press, New York, 1982).Google Scholar
[14]Kulkarni, R.P. and Limaye, B.V., “Solution of a Schrödinger equation by iterative refinement”, J. Austral. Math. Soc. (Series B) 32 (1990) 115–132.CrossRefGoogle Scholar
[15]Limaye, B.V., “Spectral perturbation and approximation with numerical experiments”, Proceedings of the Centre for Mathematical Analysis, Vol. 13 (Australian National University, 1986).Google Scholar
[16]Stetter, H., “The defect correction principle and discretization methods”, Numer. Math. 29 (1978) 425–443.CrossRefGoogle Scholar