When the eigenvalues of a given eigenvalue problem are symmetric with respect to the real and the imaginary axes, we speak about a Hamiltonian eigenvalue symmetry or a Hamiltonian structure of the spectrum. This property can be exploited for an efficient computation of the eigenvalues. For some elliptic boundary value problems it is known that the derived eigenvalue problems have this Hamiltonian symmetry. Without having a specific application in mind, we trace the question, under which assumptions the spectrum of a given quadratic eigenvalue problem possesses the Hamiltonian structure.

This paper presents the investigation of the nonlinear eigenvalue problem describing free vibrations of plates with elastically attached masses. We study properties of eigenvalues and eigenfunctions and prove the existence theorem. Theoretical results are illustrated by numerical experiments.

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of the symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to ill-conditioned nonlinear eigenvalue problems with very large sparse matrices monotone depending on the spectral parameter. To compute the smallest eigenvalue of large matrix nonlinear eigenvalue problem, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors and inner products of vectors. We investigate the convergence and derive grid-independent error estimates of these methods for computing eigenvalues. Numerical experiments demonstrate practical effectiveness of the proposed methods for a class of mechanical problems.

In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.