THE HYPERBOLIC QUADRATIC EIGENVALUE PROBLEM

To solve quadratic eigenvalue problems (QEPs), especially the gyroscopic systems, two methods are proposed: an iterative direct detection method (DDM) of the complex eigenvalues of the original QEP, and a split-linearization method (SLM) for finding the solvent matrix, which results to a standard linear eigenvalue problem (LEP) being solved to compute all eigenvalues by the symmetry extension. Reducing the dimension to one-half, the LEP is recast in a simpler QEP involving the square of the solvent. We set up two new merit functions which are minimized to detect the complex eigenvalues from the original QEP and a simpler QEP. For each eigen-parameter the merit function consists of the Euclidean norm of each derived eigen-equation, whose vector variable is solved from a derived inhomogeneous linear system. Then, the golden section search algorithm is employed to minimize the merit functions and locate the complex eigenvalue as a local minimal point. The results are compared with that computed by the cyclic-reduction-based solvent (CRS) method.

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.

A semiclassical theory of small oscillations is developed for nuclei that are subject to velocity-dependent forces in addition to the usual interatomic forces. When the velocity-dependent forces are due to a strong magnetic field, novel effects arise-for example, the coupling of vibrational, rotational, and translational modes. The theory is first developed using Newtonian mechanics and we provide a simple quantification of the coupling between these types of modes. We also discuss the mathematical structure of the problem, which turns out to be a quadratic eigenvalue problem rather than a standard eigenvalue problem. The theory is then re-derived using the Hamiltonian formalism, which brings additional insight, including a close analogy to the quantum-mechanical treatment of the problem. Finally, we provide numerical examples for the H2, HT, and HCN molecules in a strong magnetic field.

A meshless Chebyshev collocation method for eigenvalue problems of the Helmholtz equation

Linear optical properties can be accurately calculated using the Bethe-Salpeter equation. After introducing a suitable product basis for the electron-hole pairs, the Bethe-Salpeter equation is usually recast into a complex non-Hermitian eigenvalue problem that is difficult to solve using standard eigenvalue solvers. In solid-state physics, it is therefore common practice to neglect the problematic coupling between the positive- and negative-frequency branches, reducing the problem to a Hermitian eigenvalue problem [Tamm-Dancoff approximation (TDA)]. We use time-inversion symmetry to recast the full problem into a quadratic Hermitian eigenvalue problem, which can be solved routinely using standard eigenvalue solvers even at a finite wave vector $\mathbf{q}$. This allows us to access the importance of the coupling between the positive- and negative-frequency branch for prototypical solids. As a starting point for the Bethe-Salpeter calculations, we use self-consistent Green's-function methods ($\mathit{GW}$), making the present scheme entirely ab initio. We calculate the optical spectra of carbon (C), silicon (Si), lithium fluoride (LiF), and the cyclic dimer ${\mathrm{Li}}_{2}{\mathrm{F}}_{2}$ and discuss why the differences between the TDA and the full solution are tiny. However, at finite momentum transfer $\mathbf{q}$, significant differences between the TDA and our exact treatment are found. The origin of these differences is explained.

The dynamic analysis of a structural system with different material and geometric properties leads to linear eigenvalue problem such as generalized (or standard) eigenvalue problem. In general, the material and geometric properties are assumed to be in the form of crisp (or exact). However, due to several errors and insufficient or incomplete information of data, the uncertainties are assumed to be present in the material and geometric properties. These uncertain material and geometric properties may be modeled through convex normalized fuzzy sets. In standard fuzzy arithmetic, all the operands are assumed to be independent of each other. But when they are partially or completely dependent on each other, the standard fuzzy arithmetic results in a wider range. This situation is known as the “dependency problem” or “overestimation problem”. In this regard, a fuzzy-affine approach is developed to overcome the dependency problem. This proposed approach may improve the outer enclosures and give tighter bounds to the fuzzy solution set. This chapter deals with the dynamic analysis of various structural systems viz. multi degrees-of-freedom spring-mass structural system, multi-storey frame structural system, etc. by adopting the proposed approach. Several numerical examples related to the dynamic analysis of these structural systems have been worked out to illustrate the reliability and efficiency of the present approach.KeywordsAffine arithmeticFuzzy-affine arithmeticDynamic analysis of structural systemFuzzy generalized eigenvalue problemFuzzy standard eigenvalue problem

Thermal buckling response of laminated and sandwich plates using refined 2-D models

本文中我们对Helmholtz方程透射特征值问题提出一种带位移求逆的算法，此算法可以快速有效地求出任意给定的 σ 附近的几个实特征值及对应的特征向量。首先，我们用连续有限元方法对Helmholtz方程透射特征值问题进行离散，并将离散后的广义特征值问题化为一个二次特征值问题，进而对其进行线性化得到一个新的广义特征值问题。这个新的广义特征值问题排除了没有物理意义的零特征值的干扰，保留了所有的非零特征值。我们还利用位移求逆的技术，求得给定的 σ 附近的几个实特征对。我们所提出的算法对透射特征值问题的折射率没有特别的限制，即折射率可为正或负亦或是任意的实函数。最后的数值算例验证了该算法的有效性。 In this paper, we put forward a shift-and-invert algorithm to solve the transmission eigenvalue problem of Helmholtz equation, which can quickly and efficiently find out the several eigenvalues and the corresponding eigenvectors near arbitrarily given σ. First, we use the continuous finite element method to discrete the transmission eigenvalue problem of Helmholtz equation, and dis-crete the generalized eigenvalue problem into a quadratic eigenvalue problem, and then a new generalized eigenvalue problem is obtained by linearization. The new generalized eigenvalue problem eliminates the distraction of nonphysical zero eigenvalues, and preserves all the nonzero eigenvalues. Further through the use of shift-and-invert technology, we can quickly and efficiently get several real eigenpairs near given σ. The proposed algorithm has no special restrictions to the refractive index of the transmission eigenvalue problems, that is to say, the refractive index can be positive or negative or positive and negative real numbers. The final numerical example verifies the effectiveness of our algorithm.

Model updating for quadratic eigenvalue problems (QEPs) is pro- posed by Friswell, Inman and Pilkey (1998), to incorporate the measured model data into the model which only produces the mass and stifiness matrices, that closely match the experimental model data. In this paper, we consider a numer- ical model for updating QEPs which produces not only the mass and stifiness matrices, but also the damping matrix will be updated. To this end the com- plete set of eigenpairs will be employed.

Dielectric waveguide formulations usually result in generalized eigenvalue problems which are non-Hermitian in nature. We solve this class of problems by a two stage process. In the first stage, a standard lumped eigenvalue problem is solved for a few eigenpairs of interest (usually the guided modes) by Arnoldi's method accelerated by Chebyshev polynomials. The solution is then refined for the actual unlumped generalised eigenvalue problem by using inflated inverse iteration. After the eigenvalues and eigenvectors have been calculated at the first point (preferably the highest frequency), the eigenpairs for the rest of the frequencies of interest are calculated for progressively smaller frequencies by inflated inverse iteration alone, using the eigenvectors of the last step as the initial guess. To demonstrate the efficacy of this method, we use a variational formulation for anisotropic, dielectric waveguides based only on the E/sub s/ components or only on the H/sub s/ components of the electromagnetic fields that was presented by Chew and Nasir (1989). In this formulation it was shown that due of the imposition of the divergence condition the spurious waveguide modes were eliminated.

Chapter 6 - Quadratic forms and Eigenvalue problems

Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures

In a recent work (Linear Algebra Appl. 2017;520:215–241), the authors have proved that the Jacobi–Davidson iteration method attains cubic convergence rate locally for solving Hermitian standard eigenvalue problems. In this paper, we attempt to generalize these convergence results from Hermitian standard eigenvalue problems to Hermitian generalized eigenvalue problems. It indicates that the Jacobi–Davidson method at least converges linearly, and it converges quadratically and cubically if the involved correction equation is solved to the precision proportional to the norm of the current residual and to the precision proportional to the squares of the norm of the current residual, respectively. Numerical experiments are carried out to confirm these theoretical results.

SynopsisWe study the possibility of perturbing a matrix A by a diagonal matrix so that an eigenvalue problem with leading matrix A has specifiedeigenvalues when A is replaced by A+D. The particular cases presented are the one-parameter generalized eigenvalue problem (A× = λB μ×, a two-parameter eigenvalue problem (A + λB + μC)× = 0, a linked system ofsuch two-parameter problems and a quadratic eigenvalue problem (A + λB + λ2C)× = 0. The work extends results of Hadeler for the classical problem A× = λ×.

A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature