We present two novel results for small damped oscillations described by the vector differential equation Mx¨+Cx˙+Kx=0, where the mass matrix M can be singular, but standard deflation techniques ...cannot be applied. The first result is a novel formula for the solution X of the Lyapunov equation ATX+XA=−I, where A=A(v) is obtained from M,C(v)∈Rn×n, and K∈Rn×n, which are the so-called mass, damping, and stiffness matrices, respectively, and rank(M)=n−1. Here, C(v) is positive semidefinite with rank(C(v))=1. Using the obtained formula, we propose a very efficient way to compute the optimal damping matrix. The second result was obtained for a different structure, where we assume that dim(N(M))≥1 and internal damping exists (usually a small percentage of the critical damping). For this structure, we introduce a novel linearization, i.e., a novel construction of the matrix A in the Lyapunov equation ATX+XA=−I, and a novel optimization process. The proposed optimization process computes the optimal damping C(v) that minimizes a function v↦trace(ZX) (where Z is a chosen symmetric positive semidefinite matrix) using the approximation function g(v)=cv+av+bv, for the trace function f(v)≐trace(ZX(v)). Both results are illustrated with several corresponding numerical examples.
The main results of the paper are an upper and a lower bound for the Frobenius norm of the matrix sinΘ, of the sines of the canonical angles between unperturbed and perturbed eigenspaces of a regular ...generalized Hermitian eigenvalue problem Ax=λBx where A and B are Hermitian n×n matrices, under a feasible non-Hermitian perturbation. As one application of the obtained bounds we present the corresponding upper and the lower bounds for eigenspaces of a matrix pair (A,B) obtained by a linearization of regular quadratic eigenvalue problem λ2M+λD+Ku=0, where M is positive definite and D and K are semidefinite.
We also apply obtained upper and lower bounds to the important problem which considers the influence of adding a damping on mechanical systems. The new results show that for certain additional damping the upper bound can be too pessimistic, but the lower bound can reflect a behaviour of considered eigenspaces properly. The obtained results have been illustrated with several numerical examples.
This paper is concerned with the Hermitian definite generalized eigenvalue problem A-λB for block diagonal matrices A=diag(A11,A22) and B=diag(B11,B22). Both A and B are Hermitian, and B is positive ...definite. Bounds on how its eigenvalues vary when A and B are perturbed by Hermitian matrices are established. These bounds are generally of linear order with respect to the perturbations in the diagonal blocks and of quadratic order with respect to the perturbations in the off-diagonal blocks. The results for the case of no perturbations in the diagonal blocks can be used to bound the changes of eigenvalues of a Hermitian definite generalized eigenvalue problem after its off-diagonal blocks are dropped, a situation that occurs frequently in eigenvalue computations. The presented results extend those of Li and Li Linear Algebra Appl., 395 (2005), pp. 183-190. It was noted by Stewart and Sun Matrix Perturbation Theory, Academic Press, Boston, 1990 that different copies of a multiple eigenvalue may exhibit quite different sensitivities towards perturbations. We establish bounds to reflect that feature, too. We also derive quadratic eigenvalue bounds for diagonalizable non-Hermitian pencils subject to off-diagonal perturbations. PUBLICATION ABSTRACT
In this paper the damping optimization of a multi‐body oscillator with arbitrary n degrees of freedom is considered. The main contribution is a theoretical result for the optimal position of a single ...damper in a structured optimization. For that purpose the optimization criterion based on the average total energy is considered. As the damping optimization by means of the chosen optimization criterion in general is not a straight‐forward task, a numerical approach based on the formula of Veselić is introduced in this contribution. This approach can be used for efficiently calculating optimal damping for medium‐size problems with n≤2000$n \le 2000$.
The solutions of a forced gyroscopic system of ODEs may undergo large oscillations whenever some eigenvalues of the corresponding quadratic eigenvalue problem (QEP) (λ2M+λG+K)v=0,0≠v∈ℂn, are close to ...the frequency of the external force (both M,K are symmetric, M is positive definite, K is definite and G is skew-symmetric). This is the phenomenon of the so-called resonance. One way to avoid resonance is to modify some (or all) of the coefficient matrices, M, G, and K∈Rn×n in such a way that the new system has no eigenvalues close to these frequencies. This is known as the frequency isolation problem. In this paper we present frequency isolation algorithms for tridiagonal systems in which only the gyroscopic term G is modified. To derive these algorithms, the real gyroscopic QEP is first transformed into a complex hyperbolic one, which allows to translate many of the ideas in Moro and Egaña (2016) for undamped systems into the full quadratic framework. Some numerical experiments are presented.
This paper is concerned with the perturbation of a multiple eigenvalue μ of the Hermitian matrix A=diag(μI,A22) when it undergoes an off-diagonal perturbation E whose columns have widely varying ...magnitudes. When some of E’s columns are much smaller than the others, some copies of μ are much less sensitive than any existing bound suggests. We explain this phenomenon by establishing individual perturbation bounds for different copies of μ. They show that when A22-μI is definite the ith bound scales quadratically with the norm of the ith column, and in the indefinite case the bound is necessarily proportional to the product of E’s ith column norm and E’s norm. An extension to the generalized Hermitian eigenvalue problem is also presented.
We are concerned with the eigenvector-dependent nonlinear eigenvalue problem (NEPv) H(V)V=VΛ, where H(V)∈ℂn×n is a Hermitian matrix-valued function of V∈ℂn×k with orthonormal columns, i.e., VHV=Ik, ...k≤n (usually k≪n). Sufficient conditions on the solvability and solution uniqueness of NEPv are obtained, based on the well-known results from the relative perturbation theory. These results are complementary to recent ones in Cai et al. (2018), where, among others, one can find conditions for the solvability and solution uniqueness of NEPv, based on the well-known results from the absolute perturbation theory. Although the absolute perturbation theory is more versatile in applications, there are cases where the relative perturbation theory produces better results.
In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form (λ2M+λC+K)x=0, where M and K are nonsingular Hermitian ...matrices and C is a general Hermitian matrix. These results are based on new relative perturbation bounds for an equivalent regular Hermitian matrix pair A−λB. The new bounds can be applied to quadratic eigenvalue problems appearing in many relevant applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.
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 (λ2(v)M+λ(v)D(v)+K)x(v)=0, ...where M, K are positive definite Hermitian n × n matrices and D(v) is an n × n Hermitian semidefinite matrix which depends on a damping parameter vector v=v1…vk∈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 vi, 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.
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