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Backward Error Analysis for an Eigenproblem Involving Two Classes of Matrices

Published online by Cambridge University Press:  16 July 2018

Lei Zhu  and
Weiwei Xu
Lei Zhu*
Affiliation:
College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China; College of Engineering, Nanjing Agricultural University, Nanjing 210031, P. R. China.
Weiwei Xu*
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China.
*
Corresponding author. Email address:[email protected]
*Corresponding author. Email address:[email protected]
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Abstract

We consider backward errors for an eigenproblem of a class of symmetric generalised centrosymmetric matrices and skew-symmetric generalised skew-centrosymmetric matrices, which are extensions of symmetric centrosymmetric and skew-symmetric skew-centrosymmetric matrices. Explicit formulae are presented for the computable backward errors for approximate eigenpairs of these two kinds of structured matrices. Numerical examples illustrate our results.

Keywords

65F1065F15Backward errorsymmetric generalised centrosymmetric matrixskew-symmetric generalised skew-centrosymmetric matrix
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 4 , November 2014 , pp. 329 - 344
Copyright
Copyright © Global-Science Press 2014

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References

[1] Xu, Z., Zhang, K.Y. and Lu, Q., Fast Algorithms of Toeplitz Form, Northwest Industry University Press (1999).Google Scholar
[2] Good, I.J., The inverse of an entro-symmetric matrix, Technometrics, 12, 153–156 (1970).CrossRefGoogle Scholar
[3] Pye, W.C., Boullion, T.L. and Atchison, T.A., The pseudoinverse of s centro-symmetric matrix, Linear Algebra Appl. 6, 201–204 (1973).CrossRefGoogle Scholar
[4] Andrew, A., Centrosymmetric matrices, SIAM Review 40, 697–698 (1998).CrossRefGoogle Scholar
[5] Cantoni, A. and Butler, P., Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra Appl. 13, 275–288 (1976).Google Scholar
[6] Tao, D., Yasuda, M., A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices, SIAM J. Matrix Anal. Appl. 23, 885–895 (2002).CrossRefGoogle Scholar
[7] Sun, J.-G., Backward errors for the unitary eigenproblem, Technical Report UMINF-97.25, Department of Computing Science, University of Umeå, Sweden (1997).Google Scholar
[8] Bunse-Gerstner, A., Byers, R. and Mehrmann, V., A chart of numerical methods for structured eigenvalue problems, SIAM J. Matrix Anal. Appl. 13, 419–453 (1992).Google Scholar
[9] Higham, N.J., Accuracy and Stability of Numerical Algorithms, SIAM Press (2002).CrossRefGoogle Scholar
[10] Li, R.C., Relative perturbation theory: I eigenvalue and singular value variations, SIAM J. Matrix Anal. Appl. 19, 956–982 (1998).Google Scholar
[11] Li, C.K., Li, R.C. and Ye, Q. Eigenvalues of an alignment matrix in nonlinear manifold learning, Comm. Math. Sc. 5, 313–329 (2007).Google Scholar
[12] Li, R.C., Relative perturbation theory:(III) more bounds on eigenvalue variation, Linear Algebra Appl. 266, 337–345 (1997).Google Scholar
[13] Bai, Z.-J., Error analysis of Lanczos algorithm for nonsymmetric eigenvalue problem, Math. Comp. bf 62, 209–226 (1994).Google Scholar
[14] Bai, Z.-J., Day, D. and Ye, Q. ABLE: An adaptive block Lanczos method for non-Hermitian eigenvalue problems, SIAM J. Matrix Anal. Appl. 20, 1060–1082 (1999).Google Scholar
[15] Wei, M.-S., Theory and Computations for Generalised Least Squares Problems (in Chinese), Science Press, Beijing (2006).Google Scholar
[16] Sugiyama, N., Derivation of system matrices from nonlinear dynamic simulation of jet engines, Journal of Guidance, Control, and Dynamics. 17, 1320–1326 (1994).CrossRefGoogle Scholar
[17] Sugiyama, N., System identification of jet engines, J. Eng. Gas Turbines Power. 122, 19–26 (1999).Google Scholar
[18] Naderi, E., Meskin, N. and Khorasani, K., Nonlinear fault diagnosis of jet engines by using a multiple model-based approach, J. Eng. Gas Turbines Power. 134, 1–10 (2011).Google Scholar