An inverse eigenvalue problem in symmetric sparse quadratic model updating

2021 ◽  
Vol 40 (1) ◽  
Author(s):  
Suman Rakshit
Keyword(s):  
Eigenvalue Problem ◽  
Model Updating ◽  
Inverse Eigenvalue Problem ◽  
Quadratic Model ◽  
Inverse Eigenvalue

A CG-Type Method for Inverse Quadratic Eigenvalue Problems in Model Updating of Structural Dynamics

2011 ◽  
Vol 3 (1) ◽  
pp. 65-86
Author(s):  
Jiaofen Li ◽  
Xiyan Hu
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Model Updating ◽  
Inverse Eigenvalue Problem ◽  
Computationally Efficient ◽  
Type Method ◽  
Quadratic Pencil ◽  
Model Coefficient ◽  
Inverse Eigenvalue ◽  
The Difference

AbstractIn this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matricesM, DandKfor the quadratic pencilQ(λ) =λ2M+ λD+K, so thatQ(λ) has a prescribed subset of eigenvalues and eigenvectors. This method can determine the solvability of the inverse eigenvalue problem automatically. We then consider the least squares model for updating a quadratic pencilQ(λ). More precisely, we update the model coefficient matrices M, C and K so that (i) the updated model reproduces the measured data, (ii) the symmetry of the original model is preserved, and (iii) the difference between the analytical triplet (M, D, K) and the updated triplet (Mnew,Dnew,Knew) is minimized. In this paper a computationally efficient method is provided for such model updating and numerical examples are given to illustrate the effectiveness of the proposed method.

scholarly journals An Inverse Eigenvalue Problem of Hermite-Hamilton Matrices in Structural Dynamic Model Updating

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Linlin Zhao ◽  
Guoliang Chen
Keyword(s):  
Eigenvalue Problem ◽  
Generalized Inverse ◽  
Model Updating ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Structural Dynamic ◽  
Solvability Conditions ◽  
Structural Dynamic Model ◽  
Singular Value Decompositions ◽  
Inverse Eigenvalue

We first consider the following inverse eigenvalue problem: givenX∈Cn×mand a diagonal matrixΛ∈Cm×m, findn×nHermite-Hamilton matricesKandMsuch thatKX=MXΛ. We then consider an optimal approximation problem: givenn×nHermitian matricesKaandMa, find a solution(K,M)of the above inverse problem such that∥K-Ka∥2+∥M-Ma∥2=min⁡. By using the Moore-Penrose generalized inverse and the singular value decompositions, the solvability conditions and the representations of the general solution for the first problem are derived. The expression of the solution to the second problem is presented.

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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