Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be generalized to treat singular polynomial eigenvalue problems. The common denominator of all three approaches is a transformation of a singular into a regular matrix polynomial whose eigenvalues are a disjoint union of the eigenvalues of the singular polynomial, called true eigenvalues, and additional fake eigenvalues. The true eigenvalues can then be separated from the fake eigenvalues using information on the corresponding left and right eigenvectors. We illustrate the approaches on several interesting applications, including bivariate polynomial systems and ZGV points.

翻译：近来，基于秩完备摄动、秩投影或增广技术，已发展出三种计算奇异矩阵束特征值的数值方法。本文证明这三种方法均可推广至奇异多项式特征值问题的求解。三种方法的共同核心在于将奇异矩阵多项式转化为正则矩阵多项式，其特征值由原奇异多项式的特征值（称为真特征值）与新增的伪特征值构成无交并集。通过分析相应左右特征向量的信息，即可实现真特征值与伪特征值的有效分离。本文通过多个典型应用案例（包括二元多项式系统与ZGV点）对所提方法进行验证。