Abstract
The computation of the eigenvalue decomposition of symmetric matrices is one of the most investigated problems in numerical linear algebra. For a matrix of moderate size, the customary procedure is to reduce it to a symmetric tridiagonal one by means of an orthogonal similarity transformation and then compute the eigendecomposition of the tridiagonal matrix.
Recently, Malyshev and Dhillon have proposed an algorithm for deflating the tridiagonal matrix, once an eigenvalue has been computed. Starting from the aforementioned algorithm, in this manuscript we develop a procedure for computing an eigenvector of a symmetric tridiagonal matrix, once its associate eigenvalue is known.
We illustrate the behavior of the proposed method with a number of numerical examples.
The author “Nicola Mastronardi” is a member of the INdAM Research group GNCS. The scientific responsibility rests with its authors.
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Notes
- 1.
If one of the indices i, j in t i,j is either 0 or n, we set t i,j ≡ 0.
- 2.
The matrix T can be downloaded at http://users.ba.cnr.it/iac/irmanm21/TRID_SYM.
- 3.
We have used a MATLAB implementation of the MR 3 algorithm written by Petschow [13].
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Acknowledgements
The authors wish to thank the anonymous reviewers for their constructive remarks that helped improving the proposed algorithm and the presentation of the results.
The authors would like to thank Paolo Bientinesi and Matthias Petschow for providing their MATLAB implementation of the MR3 algorithm, written by Matthias Petschow.
The work of the author “Nicola Mastronardi” is partly supported by GNCS–INdAM and by CNR under the Short Term Mobility Program. The work of the author “Harold Taeter” is supported by INdAM-DP-COFUND-2015, grant number: 713485.
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Mastronardi, N., Taeter, H., Dooren, P.V. (2019). On Computing Eigenvectors of Symmetric Tridiagonal Matrices. In: Bini, D., Di Benedetto, F., Tyrtyshnikov, E., Van Barel, M. (eds) Structured Matrices in Numerical Linear Algebra. Springer INdAM Series, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-030-04088-8_9
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