SIAM Review, Volume 65, Issue 2, Page 439-470, May 2023.
Contour integral methods for eigenvalue problems seek to compute a subset of the spectrum in a bounded region of the complex plane. We briefly survey this class of algorithms, establishing a relationship to system realization and rational interpolation techniques in control theory. This connection casts contour integral methods for linear and nonlinear eigenvalue problems in a general framework that gives perspective on existing methods and suggests a broad class of new algorithms. These methods replace the usual block Hankel matrix pencils (which interpolate at infinity) with Loewner matrix pencils (enabling interpolation at many points in the complex plane). While this framework is novel for linear eigenvalue problems, we focus our presentation on the nonlinear case. The old and new methods share the same intensive computations (the solution of linear systems associated with contour integration), allowing one to explore a vast range of new eigenvalue approximations with little additional work. Numerical examples illustrate the potential of this approach. We also discuss how the concept of filter functions can be employed in this new framework, and we close with a discussion of interpolation point selection.

中文翻译：

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非线性特征值问题的轮廓积分法：系统论方法

SIAM Review，第 65 卷，第 2 期，第 439-470 页，2023 年 5 月。
特征值问题的轮廓积分方法旨在计算复平面有界区域中的频谱子集。我们简要回顾了这类算法，建立了与控制理论中系统实现和合理插值技术的关系。这种联系将轮廓积分方法用于线性和非线性特征值问题的一般框架中，该框架提供了对现有方法的看法并提出了一大类新算法。这些方法用 Loewner 矩阵笔（允许在复平面中的许多点进行插值）代替通常的块 Hankel 矩阵笔（在无穷远处插值）。虽然这个框架对于线性特征值问题是新颖的，但我们将重点放在非线性情况上。旧方法和新方法共享相同的密集计算（与轮廓积分相关的线性系统的解），允许人们探索大量新的特征值近似，而无需额外的工作。数值示例说明了这种方法的潜力。我们还讨论了如何在这个新框架中使用过滤器函数的概念，最后讨论了插值点选择。