in this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. some examples are provided to show the accuracy and reliability of the proposed method. it is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to t...

In this paper we propose numerical algorithms for solving large-scale quadratic eigenvalue problems for which a set of eigenvalues closest to a fixed target and the associated eigenvectors are of interest. The desired eigenvalues are usually with smallest modulo in the spectrum. The algorithm based on the quadratic Jacobi-Davidson (QJD) algorithm is proposed to find the first smallest eigenvalu...

In this paper, we study the quadratic tensor eigenvalue complementarity problem (QTEiCP). By a randomization process, complementarity(QC) eigenvalues are classified into two cases. For each case, QTEiCP is formulated as an equivalent generalized moment problem. The QC eigenvectors can be computed in order. Each of them solved by sequence semidefinite relaxations. We prove that such converges fi...

We introduce the quadratic two-parameter eigenvalue problem and linearize it as a singular two-parameter eigenvalue problem. This, together with an example from model updating, shows the need for numerical methods for singular two-parameter eigenvalue problems and for a better understanding of such problems. There are various numerical methods for two-parameter eigenvalue problems, but only few...

A system is defined to be an n× n matrix function L(λ) = λ2M + λD +K where M, D, K ∈ Cn×n and M is nonsingular. First, a careful review is made of the possibility of direct decoupling to a diagonal (real or complex) system by applying congruence or strict equivalence transformations to L(λ). However, the main contribution is a complete description of the much wider class of systems which can be...

We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software.

Let P, Q be compact selfadjoint operators in a Hilbert space. It is proven that the characteristic and associated vectors of the quadratic eigenvalue problem, x=\Px + (\¡X)Qx, form a Riesz basis for the cartesian product of the closure of the range of P and the closure of the range of Q. 1. Investigations in the theory of hydrodynamic stability (cf. [3, Chapter X] ; [8]) lead to the search for ...