Symmetric tridiagonal inverse quadratic eigenvalue problems with partial eigendata

Bai, Zheng-Jian

doi:10.1088/0266-5611/24/1/015005

Cited by 20 publications

(13 citation statements)

References 53 publications

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“…Note that by comparing the leading coefficients of the polynomials in ( 55)-( 56), we assert that for each j ∈ [1,4] with b j+1 > 0,…”

Section: A Proof Of Example 21mentioning

confidence: 99%

“…But when dampers are taken into consideration, the matrices associated with masses, spring stiffnesses and damping coefficients in massspring-damper systems are no longer Jacobi matrices and the quadratic inverse eigenvalue problem (QIEP) is put forth. Nevertheless, most of the literatures on mass-spring-damper systems focus on distinct eigenvalues assignment [1,3,13,16].…”

Section: Introductionmentioning

confidence: 99%

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Inverse Eigenvalue Problem For Mass-Spring-Inerter Systems

Liu¹,

Xie²,

Li³

2019

Preprint

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This paper has solved the inverse eigenvalue problem for "fixed-free" mass-chain systems with inerters. It is well known that for a spring-mass system wherein the adjacent masses are linked through a spring, the natural frequency assignment can be achieved by choosing appropriate masses and spring stiffnesses if and only if the given positive eigenvalues are distinct. However, when we involve inerters, multiple eigenvalues in the assignment are allowed. In fact, arbitrarily given a set of positive real numbers, we derive a necessary and sufficient condition on the multiplicities of these numbers, which are assigned as the natural frequencies of the concerned mass-spring-inerter system.

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“…Note that by comparing the leading coefficients of the polynomials in ( 55)-( 56), we assert that for each j ∈ [1,4] with b j+1 > 0,…”

Section: A Proof Of Example 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inverse Eigenvalue Problem For Mass-Spring-Inerter Systems

Liu¹,

Xie²,

Li³

2019

Preprint

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“…Assuming the solution of ( 1) is of the form v(t) = xe λt , using separation of variables, (1) leads to the higher order polynomial eigenvalue problem P (λ)x = 0 (2) where…”

Section: Introductionmentioning

confidence: 99%

Solution of the Linearly Structured Partial Polynomial Inverse Eigenvalue Problem

Rakshit¹,

Khare²

2019

Preprint

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In this paper, linearly structured partial polynomial inverse eigenvalue problem is considered for the n×n matrix polynomial of arbitrary degree k. Given a set of m eigenpairs (1 m kn), this problem concerns with computing the matrices A i ∈ R n×n for i = 0, 1, 2, . . . , (k − 1) of specified linear structure such that the matrix polynomial P (λ) = λ k I n + k−1 i=0 λ i A i has the given eigenpairs as its eigenvalues and eigenvectors. Many practical applications give rise to the linearly structured structured matrix polynomial. Therefore, construction of the linearly structured matrix polynomial is the most important aspect of the polynomial inverse eigenvalue problem(PIEP). In this paper, a necessary and sufficient condition for the existence of the solution of this problem is derived. Additionally, we characterize the class of all solutions to this problem by giving the explicit expressions of solutions. The results presented in this paper address some important open problems in the area of PIEP raised in De Teran, Dopico and Van Dooren [SIAM Journal on Matrix Analysis and Applications, 36(1) (2015), pp 302 − 328]. An attractive feature of our solution approach is that it does not impose any restriction on the number of eigendata for computing the solution of PIEP. The proposed method is validated with various numerical examples on a spring mass problem.

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“…Cai et al in [12] and Yuan et al in [13] deal with problems in which complete lists of eigenvalues and eigenpairs (and no definiteness constraints are imposed on M, D, K). In [14] and [15] the symmetric tridiagonal case with a partial list of eigenvalues and eigenvectors is discussed.…”

Section: Introductionmentioning

confidence: 99%

Inverse Spectral Problems for Linked Vibrating Systems and Structured Matrix Polynomials

Monfared,

Lancaster

2017

Preprint

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We show that for a given set Λ of nk distinct real numbers λ1, λ2, . . . , λ nk and k graphs on n nodes, G0, G1, . . . , G k−1 , there are real symmetric n × n matrices As, s = 0, 1, . . . , k, such that the matrix polynomial A(z) := A k z k + • • • + A1z + A0 has Λ as its spectrum, the graph of As is Gs for s = 0, 1, . . . , k − 1, and A k is an arbitrary positive definite diagonal matrix. When k = 2, this solves a physically significant inverse eigenvalue problem for linked vibrating systems (see Corollary 5.3).

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Symmetric tridiagonal inverse quadratic eigenvalue problems with partial eigendata

Cited by 20 publications

References 53 publications

Inverse Eigenvalue Problem For Mass-Spring-Inerter Systems

Inverse Eigenvalue Problem For Mass-Spring-Inerter Systems

Solution of the Linearly Structured Partial Polynomial Inverse Eigenvalue Problem

Inverse Spectral Problems for Linked Vibrating Systems and Structured Matrix Polynomials

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