Solutions of the Partially Described Inverse Quadratic Eigenvalue Problem

Kuo, Yuen-Cheng; Lin, Wen‐Wei; Xu, Shengyuan

doi:10.1137/05064134x

Cited by 40 publications

(23 citation statements)

References 17 publications

(10 reference statements)

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“…Despite the fact that some general results have been obtained for the quadratic inverse eigenvalue problems [21,23,25], we have not seen that the above-mentioned constraints being taken into account. Our contribution is innovative in that the structured QIEP is cast as a system of inequalities whose solvability can then be checked numerically.…”

Section: Introduction the Time-invariant Second Order Differential Smentioning

confidence: 99%

Structured Quadratic Inverse Eigenvalue Problem, I. Serially Linked Systems

Chu¹,

Buono²,

Yu³

2007

SIAM J. Sci. Comput.

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Abstract. Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constraints must be respected. Consequently, they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation. Knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications. However, the issue of solvability is problem dependent and has to be addressed structure by structure. This paper considers one particular structure where the elements of the physical system, if modeled as a mass-spring system, are serially linked. The discussion recasts both undamped or damped problems in a framework of inequality systems that can be adapted for numerical computation. Some open questions are described.

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Section: Introduction the Time-invariant Second Order Differential Smentioning

confidence: 99%

Structured Quadratic Inverse Eigenvalue Problem, I. Serially Linked Systems

Chu¹,

Buono²,

Yu³

2007

SIAM J. Sci. Comput.

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“…It is intuitively true that if the number k of prescribed eigenpair is capped by the bound k < 3n 1=2 (27) then the system (26) is underdetermined and the solutions form a subspace of dimensionality 3nn 1=2 nk; otherwise, the algebraic system and, hence, the QIEP, will have only a trivial solution. In what follows, we prove that this conjecture is indeed true.…”

Section: Quadratic Inverse Eigenvalue Problemmentioning

confidence: 99%

“…Because the entire A and part of B are free, there is room to impose additional eigeninformation to the pencil. In [27], for instance, it was argued that additional n eigenvalues could be specified. In our context, we ask how many more eigenpairs can be prescribed.…”

Section: A Case K Nmentioning

confidence: 99%

Spillover Phenomenon in Quadratic Model Updating

et al. 2008

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Model updating Q2concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is often desirable to match only the part of observed data without tampering with the other part of unmeasured or unknown eigenstructure inherent in the original model. Such an updating, if possible, is said to have no spillover. This paper studies the spillover phenomenon in the updating of quadratic pencils. In particular, it is shown that an updating with no spillover is always possible for undamped quadratic pencils, whereas spillover for damped quadratic pencils is generally unpreventable.

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“…There is already a long list of studies on this subject. See, for example, [2,16,17,19,20,21,22,25,27] and the references contained therein. In this demonstration, we consider the special QIEP where the entire eigeninformation is given:…”

Section: Applicationsmentioning

confidence: 99%

Spectral decomposition of real symmetric quadratic $\lambda $-matrices and its applications

Chu

2009

Math. Comp.

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Abstract. Spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions. The canonical form often facilitates discussions which, otherwise, would be complicated and involved. Spectral decomposition is of fundamental importance in many applications. The well-known GLR theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees, but its development is based on complex numbers. This paper modifies the GLR theory for the special application to real symmetric quadratic matrix polynomials, Q(λ) = Mλ 2 + Cλ + K, M nonsingular, subject to the specific restriction that all matrices in the representation be real-valued. It is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which, in turn, can be constructed from the spectral data. Applications to a variety of challenging inverse problems are discussed.

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Solutions of the Partially Described Inverse Quadratic Eigenvalue Problem

Cited by 40 publications

References 17 publications

Structured Quadratic Inverse Eigenvalue Problem, I. Serially Linked Systems

Structured Quadratic Inverse Eigenvalue Problem, I. Serially Linked Systems

Spillover Phenomenon in Quadratic Model Updating

Spectral decomposition of real symmetric quadratic $\lambda $-matrices and its applications

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