In this paper, we propose a unified approach for solving structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomials that include: symmetric, Hermitian, $\star$-even and $\star$-odd quadratic matrix polynomials. Finally, we show that the obtained analytical expressions of perturbations can realize existing results for structured polynomials that arise in real-world applications, as special cases. The obtained results are supported with numerical examples.
In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $λ^2 M x + λC x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices and $C$ is a general Hermitian matrix. We base our findings on new results for an equivalent regular Hermitian matrix pair $A-λB$. The new bounds can be applied to many interesting quadratic eigenvalue problems appearing in applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.
The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $δ$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and …
Three new Arnoldi-type methods are presented to accelerate the modal analysis and critical speed analysis of the damped rotor dynamics finite element (FE) model. They are the linearized quadratic eigenvalue problem (QEP) Arnoldi method, the QEP Arnoldi method, and the truncated generalized standard eigenvalue problem (SEP) Arnoldi method. And, they correspond to three reduction subspaces, including the linearized QEP Krylov subspace, the QEP Krylov subspace, and the truncated generalized SEP Krylov subspace, where the first subspace is also used in the existing Arnoldi-type methods. The numeri…
We consider a new algorithm in light of the min-max Collatz-Wielandt formalism to compute the principal eigenvalue and the eigenvector (eigen-function) for a class of positive Perron-Frobenius-like operators. Such operators are natural generalizations of the usual nonnegative primitive matrices. These have nontrivial applications in PDE problems such as computing the principal eigenvalue of Dirichlet Laplacian operators on general domains. We rigorously prove that for general initial data the corresponding numerical iterates converge globally to the unique principal eigenvalue with quadratic c…
Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems
IIT Guwahati（印度理工学院古瓦哈提分校）；TU Berlin（柏林工业大学）；Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics（奥西耶克约瑟普尤拉伊斯特罗斯梅亚耶尔大学应用数学与信息学院）
On Some Inverse Eigenvalue Problems of Quadratic Palindromic Systems
LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Sciences, Beijing University of Posts and Telecommunications（北京邮电大学理学院）
Certain applications that analyze damping effects require the solution of quadratic eigenvalue problems (QEPs). We use refined isogeometric analysis (rIGA) to solve quadratic eigenproblems. rIGA discretization, while conserving desirable properties of maximum-continuity isogeometric analysis (IGA), reduces the interconnection between degrees of freedom by adding low-continuity basis functions. This connectivity reduction in rIGA's algebraic system results in faster matrix LU factorizations when using multifrontal direct solvers. We compare computational costs of rIGA versus those of IGA when e…
Generalized finite element methods for quadratic eigenvalue problems
Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学）；Insitute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn（波恩莱茵弗里德里希·威廉姆斯大学数值模拟研究所）
Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose spectral problem can be written as the eigenvalue problem of some holomophic Fredholm operator function. Focusing on the finite element methods (conforming, discontinuous Galerkin, etc.), we show that the regular convergence of discrete holomorphic operator functions follows from the approximation property of the finite element spaces and the compact converg…
In this paper, a piecewise quadratic finite element method on rectangular grids for the $H^1$ problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is $O(h^2)$ in the energy norm on uniform grids. Besides, a lower bound of the $L^2$-norm error is also proved, which makes the capacity analysis of this scheme more clear. On the other hand, for the eigenvalue problem, the numerical eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are present…
We prove convergence of the Maxwell eigenvalue problem using quadratic or higher Lagrange finite elements on Worsey-Farin splits in three dimensions. To do this, we construct two Fortin-like operators to prove uniform convergence of the corresponding source problem. We present numerical experiments to illustrate the theoretical results.
This paper is devoted to the computation of transmission eigenvalues in the inverse acoustic scattering theory. This problem is first reformulated as a two by two boundary system of boundary integral equations. Next, utilizing the Schur complement technique, we develop a Schur complement operator with regularization to obtain a reduced system of boundary integral equations. The Nyström discretization is then used to obtain an eigenvalue problem for a matrix. We employ the recursive integral method for the numerical computation of the matrix eigenvalue. Numerical results show that the proposed …
