School of Computer Science and Communication
A common approach to study properties of problems appearing in science and engineering is to solve or compute some relevant solutions of an associated eigenvalue problem. In these situations, it is very desirable to use physical models with high accuracy, which typically lead to eigenvalue problems with either large size and/or some form of nonlinearity and particular structure. We here present a review of numerical methods for some common generalizations of the eigenvalue problem including (what is commonly called) the nonlinear eigenvalue problem. We discuss nonlinear eigenvalue problems arising in the study of delay-differential equations, two-parameter eigenvalue problems, quadratic eigenvalue problems and eigenvalue problems with Kronecker structure. Moreover, we discuss iterative numerical methods including Newton-like methods and methods based on the Arnoldi method. We finally mention some recent research directions related to a different type of eigenvalue problem appearing in the study of the Gross-Pitaevskii equation.