This article is devoted to the numerical solution of large-scale quadratic eigenvalue problems. Such problems arise in a wide variety of applications, such as the dynamic analysis of structural mechanical systems, acoustic systems, fluid mechanics, and signal processing. We first introduce a generalized second-order Krylov subspace based on a pair of square matrices and two initial vectors and ...

A restarted Arnoldi algorithm is given that computes eigenvalues and eigenvectors. It is related to implicitly restarted Arnoldi, but has a simpler restarting approach. Harmonic and regular RayleighRitz versions are possible. For multiple eigenvalues, an approach is proposed that first computes eigenvalues with the new harmonic restarted Arnoldi algorithm, then uses random restarts to determine...

For real symmetric eigenvalue problems, there are a number of algorithms that are mathematically equivalent, for example, the Lanczos algorithm, the Arnoldi method and the unpreconditioned Davidson method. The Lanczos algorithm is often preferred because it uses signiicantly fewer arithmetic operations per iteration. To limit the maximum memory usage, these algorithms are often restarted. In re...

The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. Our first important result is a characterization of a general nonlinear eigenvalue problem (NEP) as a standard but infinite dimensional eigenvalue problem involving an integration operator denoted B. In this paper we present a new algorithm equi...

In this paper, we propose a multi-level finite element method for the transmission eigenvalue problem of anisotropic media. The problem is non-standard and non-self-adjoint with important applications in inverse scattering theory. We employ a suitable finite element method to discretize the problem. The resulting generalized matrix eigenvalue problem is large, sparse and non-Hermitian. To compu...

Computing the eigenvalues and eigenvectors of a large sparse nonsymmetric matrix arises in many applications and can be a very computationally challenging problem. In this paper we propose the Augmented Block Householder Arnoldi (ABHA) method that combines the advantages of a block routine with an augmented Krylov routine. A public domain MATLAB code ahbeigs has been developed and numerical exp...

The Davidson method is a popular preconditioned variant of the Arnoldi method for solving large eigenvalue problems. For theoretical, as well as practical reasons the two methods are often used with restarting. Frequently, information is saved through approximated eigen-vectors to compensate for the convergence impairment caused by restarting. We call this scheme of retaining more eigenvectors ...

When using the Arnoldi method for approximating f(A)b, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which was universally applicable, efficient, and stable at the same time. We ut...