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.

中文翻译：

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非线性特征值问题

非线性特征值问题出现在各种科学和工程应用中，在过去十年中，数值方法的发展取得了许多突破。本文调查了与非线性地依赖于单个标量参数的矩阵值函数相关的非线性特征值问题，特别强调了它们的数学特性和可用的数值求解技术。回顾了基于牛顿法、轮廓积分和有理插值采样的求解器。为每个求解器类别选择适当参数的问题通过数值示例进行了讨论和说明。该调查还包含大量 MATLAB 代码片段，可用于对所讨论的方法进行交互式探索。