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Some Refined Eigenvalue Perturbation Bounds for Two-by-Two Block Hermitian Matrices

Published online by Cambridge University Press:  28 May 2015

Xianping Wu ,
Wen Li  and
Xiaofei Peng
Xianping Wu*
Affiliation:
Department of Mathematics, School of Basic Courses, Guangdong Pharmaceutical University, Guangzhou, 510006, P. R., China
Wen Li
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R., China
Xiaofei Peng
Affiliation:
School of Software, South China Normal University, Foshan, 528225, P. R., China
*
*Corresponding author. Email addresses: [email protected] (X. Wu), [email protected] (W. Li), [email protected] (X. Peng)
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Abstract

We consider eigenvalue perturbation bounds for Hermitian matrices, which are associated with problems arising in various computational science and engineering applications. New bounds are discussed that are sharper than some existing ones, including the well-known Weyl bound. Two numerical examples are investigated, to illustrate our theoretical presentation.

Keywords

65F10Eigenvalue perturbationHermitian block matrixWeyl bound
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 2 , May 2015 , pp. 126 - 137
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Bai, Z.-Z., Ng, M.K. and Wang, Z.-Q., Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl. 31, 410–433 (2009).Google Scholar
[2]Bai, Z.-Z. and Li, G.-Q., Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal. 23, 561–580 (2003).Google Scholar
[3]Benzi, M., Golub, G.H. and Liesen, J., Numerical solution of saddle point problems, Acta Numerica 14, 1–137 (2005).Google Scholar
[4]Cheng, G.H., Tan, Q. and Wang, Z.D., A note on eigenvalues of perturbed 2 × 2 block Hermitian matrices, Linear and Multilinear Algebra 63, 820–825 (2015).Google Scholar
[5]Elman, H.C., Ramage, A. and Silvester, D.J., Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw. 33, 251–268 (2007).Google Scholar
[6]Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge University Press (1985).Google Scholar
[7]Gould, N.I.M. and Simoncini, V., Spectral analysis of saddle point matrices with indefinite leading blocks, SIAM J. Matrix Anal. Appl. 31, 1152–1171 (2009).Google Scholar
[8]Li, C.-K. and Li, R.-C., A note on eigenvalues of perturbed Hermitian matrices, Linear Algebra Appl. 395, 183–190 (2005).Google Scholar
[9]Li, W., Vong, S.W. and Peng, X.F., On eigenvalue perturbation bounds for Hermitian block tridiag-onal matrices, Appl. Num. Math. 83, 38–50 (2014).Google Scholar
[10]Mathias, R., Quadratic residual bounds for the Hermitian eigenvalue problem, SIAM J. Matrix Anal. Appl. 19, 541–550 (1998).Google Scholar
[11]Nakatsukasa, Y., Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices, Appl. Num. Math. 62, 67–78 (2012).Google Scholar