On the Geršgorin-type localizations for nonlinear eigenvalue problems

Abstract

Since nonlinear eigenvalue problems appear in many applications, the research on their proper treatment has drawn a lot of attention lately. Therefore, there is a need to develop computationally inexpensive ways to localize eigenvalues of nonlinear matrix-valued functions in the complex plane, especially eigenvalues of quadratic matrix polynomials. Recently, few variants of the Geršgorin localization set for more general eigenvalue problems, matrix pencils and nonlinear ones, were developed and investigated. Here, we introduce a more general approach to Geršgorin-type sets for nonlinear case using diagonal dominance, prove some properties of such sets and show how they perform on several problems in engineering.

Introduction

Nonlinear eigenvalue problems (NLEPs) occur in many applications in modern science, for example in models with delay or radiation, when applying transform methods for analysis of differential and difference equations, to name some of them, [1]. The simplest nonlinear eigenvalue problem is a quadratic eigenvalue problem (QEP) that consists of finding scalars λ and nonzero vectors x and y satisfying

$$({\lambda }^{2}C+\lambda B+A)x=0,y^{*}({\lambda }^{2}C+\lambda B+A)=0$$

where A, B, C are n × n complex matrices and x and y are the right and left eigenvectors, respectively, corresponding to the eigenvalue λ.

QEPs are an important class of nonlinear eigenvalue problems that are more challenging to solve than the standard eigenvalue problems (SEPs) and generalized eigenvalue problems (GEPs). Nonetheless, for (dense) medium size matrices there is a good and robust software to solve QEPs, namely the quadeig algorithm by Hammerling et al. [4]. A wide range of practical problems that can be formulated as QEP in various disciplines (dynamical analysis of structural mechanical and acoustic systems, gyroscopic systems, electrical circuit simulation, fluid mechanics, modeling microelectronic mechanical systems) present a good motivation for its detailed treatment, see [1].

A standard way to threat a QEP is by using a linearization, i.e., to equivalently represent it as GEP in higher dimensions, and then solve with GEP solvers. Since the matrix structure plays a crucial role in numerical methods for computing (generalized) eigenvalues, a lot of progress has been done in constructing linearizations that produce GEP’s that inherit the structure of the original QEP’s, [5], [11], [12], [14].

However, in many applications the suitable structure (typically symmetric/ skew-symmetric, positive definite, palindromic...) of the quadratic matrix polynomial can be absent, and, therefore, one would either need to use standard linearizations which may involve the inverse of the leading matrix C, some more computationally demanding algorithms or use shift-and-inverse type methods and work directly with QEP. In any of these cases, especially for large and sparse problems, one would be interested to develop computationally cheap ways that reveal at least some of the properties of the spectra of such a quadratic matrix polynomial without use of numerically expensive matrix factorizations. In case of shift-and-invert methods, such area in the complex plane where the eigenvalues are located is essential for the choice of shifts. Desirably, computational cost of such a (crude) method that localizes quadratic eigenvalues should be of the order $\mathcal{O}\left(n\right),$ and in some cases can be of order $\mathcal{O}\left(n^2\right)$.

A typical way to localize matrix spectra is by application of the famous Geršgorin’s theorem and its generalizations. While for SEP extensive literature on such localization methods exists, for example see [6], [15], GEP has just recently been considered in greater detail, [9], [10], [13], while the Geršgorin’s theorem of NLEP was introduced in [2].

Here, we consider several Geršgorin-type sets for NLEP using the approach suggested in [9], [10] for the case of GEPs, investigate some of their properties (compactness, isolation) and illustrate their use on several problems that arise in applications. As a consequence we generalize the concept of NLEP Geršgorin set obtained in [2].

In Section 2 we introduce the framework of diagonal dominant matrices, and in Section 3 we construct the Geršgorin-type localization sets for NLEP, and, more specifically, QEP. Finally, in Section 4 we illustrate the use of these sets for some problems from the collection [1]: acoustic wave 1d, bilby and wiresaw 1 for QEP, and Hadeler and time-delay for nonquadratic NLEP.