in this paper, we represent an inexact inverse subspace iteration method for com- puting a few eigenpairs of the generalized eigenvalue problem ax = bx[q. ye and p. zhang, inexact inverse subspace iteration for generalized eigenvalue problems, linear algebra and its application, 434 (2011) 1697-1715 ]. in particular, the linear convergence property of the inverse subspace iteration is preserved.

For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous  differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.

Let G be a simple graph with vertex set V (G) = {1, 2, . . . , n} and (0, 1)adjacency matrix A. The eigenvalue μ of A is said to be a main eigenvalue of G if the eigenspace E(μ) is not orthogonal to the all-1 vector j. An eigenvector x is a main eigenvector if xj 6= 0. The main eigenvalues of the connected graphs of order ≤ 5 are listed in [12, Appendix B], and those of all the connected graphs...

The Dirichlet eigenvalue or “drum” problem in a domain Ω ⊂ R2 becomes numerically challenging at high eigenvalue (frequency) E. In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes. It requires a number of degrees of freedom scaling as √ E, the number of wavelengths on the boundary, in contrast to direct discretization for which this scaling is ...

Time-delays are important components of many dynamical systems that describe coupling or interconnection between dynamics, propagation or transport phenomena, and heredity and competition in population dynamics. The stabilization with time delay in observation or control represents difficult mathematical challenges in the control of distributed parameter systems. It is well-known that the stabi...

We consider the asymptotic stability of transition front solutions for Cahn–Hilliard systems on R. Such equations arise naturally in the study of phase separation, and systems describe cases in which three or more phases are possible. When a Cahn–Hilliard system is linearized about a transition front solution, the linearized operator has an eigenvalue at 0 (due to shift invariance), which is no...

In this paper we study constrained eigenvalue optimization of noncommutative (nc) polynomials, focusing on the polydisc and the ball. Our three main results are as follows: (1) an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition; (2) (eigenvalue) optima for nc polynomials can be computed using a single semidefinite program (SDP) – this shar...

In this article, we study, as the coefficient s → ∞, the asymptotic behavior of the principal eigenvalue of the eigenvalue problem −φ′′(x)− 2sm′(x)φ′(x) + c(x)φ(x) = λ(s)φ(x), 0 < x < 1, complemented by a general boundary condition. This problem is relevant to nonlinear propagation phenomena in reaction-diffusion equations. The main point is that the advection (or drift) term m allows natural d...

We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The pr...

This report describes an algorithm for the efficient computation of several extreme eigenvalues and corresponding eigenvectors of a large-scale standard or generalized real symmetric or complex Hermitian eigenvalue problem. The main features are: (i) a new conjugate gradient scheme specifically designed for eigenvalue computation; (ii) the use of the preconditioning as a cheaper alternative to ...