During the last years nonlinear eigenvalue problems of the type T (λ)x = 0 became more and more important in many applications with the rapid development of computing performance. In parallel the minmax theory for symmetric nonlinear eigenvalue problems was developed which reveals a strong connection to corresponding linear problems. We want to review the current theory for symmetric nonlinear ...

This paper is concerned with the existence, nonexistence, uniqueness, and multiplicity of positive solutions for a class of eigenvalue problems of nonlinear fractional differential equations with a nonlinear integral term and a disturbance parameter in the boundary conditions. By using fixed point index theory we give the critical curve of eigenvalue λ and disturbance parameterμ that divides th...

The FEAST algorithm [5] is an algorithm aimed at solving generalized eigenvalue problems AX = BXΛ, where A,B are n×n matrices. In [4] we presented a short analysis of FEAST involving this eigenvalue problem. In the present work we focus on the real symmetric eigenvalue problem AX = XΛ, where A = A, XX = I and Λ is a diagonal matrix consisting of real eigenvalues. These eigenvalues are sought in...

An inverse eigenvalue problem approach to system design is considered. The Cayley-Hamilton theorem is developed for the general case involving the generalized eigenvalue vibration problem. Since many solutions exist for a desired frequency spectrum, a discussion of the required design information and suggestions for including structural constraints are given. An algorithm for solving the invers...

In this paper, a local and parallel algorithm based on the multilevel discretization is proposed for solving the eigenvalue problem by the finite element method. With this new scheme, the eigenvalue problem solving in the finest grid is transferred to solutions of the eigenvalue problems on the coarsest mesh and a series of solutions of boundary value problems on each level mesh. Therefore this...

The class of nonsingular connected weighted directed graphs with an unweighted undirected branch is considered in this article. This paper investigates the monotonicity properties of the first eigenvectors of such graphs along certain paths. The paper describes how the first eigenvalue of such graphs changes under some perturbation. It is shown that replacing a branch which is a tree by a path ...

Two different Lanczos-type methods for the linear response eigenvalue problem are analyzed. The first one is a natural extension of the classical Lanczos method for the symmetric eigenvalue problem while the second one was recently proposed by Tsiper specially for the linear response eigenvalue problem. Our analysis leads to bounds on errors for eigenvalue and eigenvector approximations by the ...

In earlier papers, the Bauer–Fike technique was applied to the ordinary eigenvalue problem Ax = λx, the generalized eigenvalue problem Ax = λBx and the matrix polynomial eigenvalue problem ∑m k=0 λAkx = 0. General multiple eigenvalues were dealt with and condition numbers were obtained for individual as well as clusters of eigenvalues. In this paper, we shall generalize the technique to the eig...

The class of connected 3-colored digraphs containing exactly one nonsingular cycle is considered in this article. The main objective is to study the smallest Laplacian eigenvalue and the corresponding eigenvectors of such graphs. It is shown that the smallest Laplacian eigenvalue of such a graph can be realized as the algebraic connectivity (second smallest Laplacian eigenvalue) of a suitable u...