On the Geršgorin-type localizations for nonlinear eigenvalue problems

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.