We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such "migrating" eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.

翻译：我们描述了一种求解一般参数（非线性）特征值问题的新型算法。该方法包含两个步骤：首先，在参数的某些取值处收集非参数版本问题的高精度解；其次，将这些解组合以获得参数特征值的全局近似。为收集非参数数据，我们采用基于非侵入式轮廓积分的方法，但该方法无法追踪随参数变化而移入/移出轮廓的特征值。针对此类"迁移"特征值仅具有部分信息的情况，我们描述了执行跨参数组合步骤的特殊策略。此外，我们特别关注了发生分岔的特征值近似问题。最后，我们提出一种自适应策略，即使对目标特征值的行为没有任何先验信息，也能有效应用该方法。数值实验表明，该算法能够实现非常高的逼近精度。