We present Gerschgorin-type eigenvalue inclusion sets applicable to generalized eigenvalue problems. Our sets are defined by circles in the complex plane in the standard Euclidean metric, and are easier to compute than known similar results. As one application we use our results to provide a forward error analysis for a computed eigenvalue of a diagonalizable pencil.

This paper concerns with the sensitivity analysis for the multivariate eigenvalue problem (MEP). The concept of a simple multivariate eigenvalue of a matrix is generalized to the MEP and the first-order perturbation expansions of a simple multivariate eigenvalue and the corresponding multivariate eigenvector are presented. The explicit expressions of condition numbers, perturbation upper bounds...

Perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil A-ÀB with real spectrum are developed. It is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices. The applications of these bounds to the spectral variation...

In the past two decades, due to study on matrix theory and some engineering background problems, many scholars dedicated to special matrix , and obtained some important and valuable results (TingZhu, 2007 Yigeng Huang, 1994). But in combination matrix theory, combinatorics, probability theory (especially Markov chain), mathematical economics and reliability theory etc. areas there is a special ...

A modal decomposition strategy based on state-variable ensembles is formulated. A nonsymmetric, generalized eigenvalue problem is constructed. The data-based eigenvalue problem is related to the generalized eigenvalue problem associated with free-vibration solutions of the state-variable formulation of linear multi-degree-of-freedom systems. For linear free-response data, the inverse-transpose ...

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing nonorthogonal basis functions for ab initio electronic structure calculations. In this paper, we propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative...

Eigenvalue analysis plays an important role in understanding physical phenomena. However, if one deals with strongly nonnormal matrices or operators, the eigen-values alone may not tell the full story. A popular tool which can be useful to get more insight in the reliability or sensitivity of eigenvalues is "-pseudospectra. Apart from "-pseudospectra we consider other tools which might help to ...

Iterative methods for solving large, sparse, symmetric eigenvalue problems often encounter convergence diiculties because of ill-conditioning. The Generalized Davidson method is a well known technique which uses eigenvalue preconditioning to surmount these diiculties. Preconditioning the eigenvalue problem entails more subtleties than for linear systems. In addition, the use of an accurate conv...

We address the count of isolated and embedded eigenvalues in a generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines spectral stability of nonlinear waves in a Hamiltonian dynamical system. The theory is based on the Pontryagin’s Invariant Subspace theorem...