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General Solutions for a Class of Inverse Quadratic Eigenvalue Problems

Published online by Cambridge University Press:  28 May 2015

Xiaoqin Tan  and
Li Wang
Xiaoqin Tan*
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Li Wang*
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

Based on various matrix decompositions, we compare different techniques for solving the inverse quadratic eigenvalue problem, where n × n real symmetric matrices M, C and K are constructed so that the quadratic pencil Q(λ) = λ2M + λC + K yields good approximations for the given k eigenpairs. We discuss the case where M is positive definite for 1 ≤ k ≤ n, and a general solution to this problem for n + 1 ≤ k ≤ 2n. The efficiency of our methods is illustrated by some numerical experiments.

Keywords

65F18Quadratic eigenvalue probleminverse quadratic eigenvalue problempartially prescribed spectral information
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 1 , February 2014 , pp. 69 - 81
Copyright
Copyright © Global-Science Press 2014

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References

[1]Alvin, K.F. and Park, K.C., Second-order structural identification procedure via state-space-based system identification, AIAAJ. 32, 397–406 (1994).CrossRefGoogle Scholar
[2]Cai, Y-F., Kuo, Y-C., Lin, W-W., and Xu, S-F., Solution to a quadratic inverse eigenvalue problem, Linear Algebra Appl. 430, 1590–1606 (2009).Google Scholar
[3]Chu, M.T., Inverse eigenvalue problems, SIAM Rev. 40, 1–39 (1998).Google Scholar
[4]Chu, M.T. and Golub, G.H., Structured quadratic inverse eigenvalue problem, Acta Numer. 11, 1–71 (2002).Google Scholar
[5]Dai, H. and Bai, Z.-Z., On smooth LU decompositions with applications to solutions of nonlinear eigenvalue problems, J. Comput. Math. 28, 745–766 (2010).Google Scholar
[6]Datta, B.N., Elhay, S., Ram, Y.M., and Sarkissian, D.R., Partial eigenstructure assignment for the quadratic pencil, J. Sound Vibration 230, 101–110 (2000).Google Scholar
[7]Friswell, M.I. and Mottershead, J.E., Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht., 1995.CrossRefGoogle Scholar
[8]Gohberg, I., Lancaster, P., and Rodman, L., Spectral analysis of self adjoint matrix polynomials, Ann. Math. 112, 33–71 (1980).Google Scholar
[9]Joseph, K.T., Inverse eigenvalue problem in structral design, AIAAJ. 30, 2890–2896 (1992).CrossRefGoogle Scholar
[10]Kuo, Y.-C., Lin, W-W., and Xu, S-F., Solutions of the partially described inverse quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl. 29, 33–53 (2006).CrossRefGoogle Scholar
[11]Lancaster, P. and Maroulas, J., Inverse eigenvalue problems for damped vibrating system, J. Math Anal. Appl. 123, 238–361 (1987).Google Scholar
[12]Lancaster, P. and Prells, U., Inverse Problems for damped vibrating system, J. Sound Vibration 283, 891–914 (2005).Google Scholar
[13]Lancaster, P., Inverse spectral problems for semisimple damped vibrating systems, SIAM J. Matrix Anal. Appl. 29, 279–301 (2007).CrossRefGoogle Scholar
[14]Liao, A.-P. and Bai, Z.-Z., The constrained solutions of two matrix equations, Acta Math. Sin. 18, 671–678 (2002).Google Scholar
[15]Liao, A.-P. and Bai, Z.-Z., Least-squares solution of AXB = D over symmetric positive semidefinite matrices, J. Comput. Math. 21, 175–182 (2003).Google Scholar
[16]Liao, A.-P., Bai, Z.-Z., and Lei, Y., Best approximate solution of matrix equation AXB + CYD = E, SIAM J. Matrix Anal. Appl. 27, 675–688 (2005).CrossRefGoogle Scholar
[17]Ram, Y.M. and Elhay, S., An inverse eigenvalue problem for the symmetric tridiagonal quadratic pencil with application of damped oscillatory systems, SIAM J. Appl. Math. 56, 232–244 (1996).CrossRefGoogle Scholar
[18]Yuan, Y. and Dai, H., On a class of inverse quadratic eigenvalue problem, Linear Algebra Appl. 235, 2662–2669 (2011).Google Scholar
[19]Yuan, Y. and Dai, H., Solution of an inverse monic quadratic eigenvalue problem, Linear Algebra Appl. 434, 2367–2381 (2011).Google Scholar