Truhar, Ninoslav; Tomljanović, Zoran; Puvača, Matea Approximation of damped quadratic eigenvalue problem by dimension reduction. (English) Zbl 1428.15015 Appl. Math. Comput. 347, 40-53 (2019). Summary: This paper presents an approach to the efficient calculation of all or just one important part of the eigenvalues of the parameter dependent quadratic eigenvalue problem \((\lambda^2(\mathbf{v}) M + \lambda(\mathbf{v}) D(\mathbf{v}) + K) x(\mathbf{v}) = 0\), where \(M, K\) are positive definite Hermitian \(n\times n\) matrices and \(D(\mathbf{v})\) is an \(n\times n\) Hermitian semidefinite matrix which depends on a damping parameter vector \(\mathbf{v} = [v_1 \dots v_k ] \in \mathbb{R}_+^k\). With the new approach one can efficiently (and accurately enough) calculate all (or just part of the) eigenvalues even for the case when the parameters \(v_i\), which in this paper represent damping viscosities, are of the modest magnitude. Moreover, we derive two types of approximations with corresponding error bounds. The quality of error bounds as well as the performance of the achieved eigenvalue tracking are illustrated in several numerical experiments. Cited in 2 Documents MSC: 15A24 Matrix equations and identities 15A18 Eigenvalues, singular values, and eigenvectors 70J10 Modal analysis in linear vibration theory 65F15 Numerical computation of eigenvalues and eigenvectors of matrices Keywords:dimension reduction; parameter dependent eigenvalue problem; tracking eigenvalues; eigenvalue error bounds PDFBibTeX XMLCite \textit{N. Truhar} et al., Appl. Math. Comput. 347, 40--53 (2019; Zbl 1428.15015) Full Text: DOI References: [1] Truhar, N.; Veselić, K., An efficient method for estimating the optimal dampers’ viscosity for linear vibrating systems using Lyapunov equation, SIAM J. Matrix Anal. 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