The nonlinear eigenvalue problem

Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs
in the development of numerical methods. This article surveys nonlinear
eigenvalue problems associated with matrix-valued functions which depend
nonlinearly on a single scalar parameter, with a particular emphasis on their
mathematical properties and available numerical solution techniques. Solvers
based on Newton's method, contour integration and sampling via rational
interpolation are reviewed. Problems of selecting the appropriate parameters
for each of the solver classes are discussed and illustrated with numerical
examples. This survey also contains numerous MATLAB code snippets that
can be used for interactive exploration of the discussed methods.