We consider nonlinear eigenvalue problems to compute all eigenvalues in a bounded region on the complex plane. Based on domain decomposition and contour integrals, two robust and scalable parallel multi-step methods are proposed. The first method 1) uses the spectral indicator method to find eigenvalues and 2) calls a linear eigensolver to compute the associated eigenvectors. The second method 1) divides the region into subregions and uses the spectral indicator method to decide candidate regions that contain eigenvalues, 2) computes eigenvalues in each candidate subregion using Beyn's method; and 3) verifies each eigenvalue by substituting it back to the system and computes the smallest eigenvalue. Each step of the two methods is carried out in parallel. Both methods are robust, accurate, and does not require prior knowledge of the number and distribution of the eigenvalues in the region. Examples are presented to show the performance of the two methods.

翻译：本文考虑在复平面有界区域内计算所有特征值的非线性特征值问题。基于区域分解与等高线积分技术，提出两种稳健可扩展的并行多步方法。第一种方法：1）使用谱指示法定位特征值；2）调用线性特征值求解器计算对应特征向量。第二种方法：1）将区域划分为子区域，通过谱指示法判定包含特征值的候选子区域；2）采用Beyn方法计算每个候选子区域内的特征值；3）通过将特征值代入原系统并求取最小特征值进行验证。两种方法的每个步骤均以并行方式执行。这两种方法兼具稳健性与精确性，且无需预知区域内特征值的数量与分布先验信息。最后通过算例展示两种方法的性能表现。