This article presents a new method for computing all eigenvalues and eigenvectors of quadratic matrix pencil Q (λ)=λ 2 M + λ C + K . It is an upgrade of the quadeig algorithm by Hammarlinget al., which attempts to reveal and remove by deflation a certain number of zero and infinite eigenvalues before QZ iterations. Proposed modifications of the quadeig framework are designed to enhance backward stability and to make the process of deflating infinite and zero eigenvalues more numerically robust. In particular, careful preprocessing allows scaling invariant/component-wise backward error and thus a better condition number. Further, using an upper triangular version of the Kronecker canonical form enables deflating additional infinite eigenvalues, in addition to those inferred from the rank of M . Theoretical analysis and empirical evidence from thorough testing of the software implementation confirm superior numerical performances of the proposed method.

中文翻译：

---

无限和零特征值紧缩的新数值算法及二次特征值问题的全解

本文提出了一种计算二次矩阵铅笔的所有特征值和特征向量的新方法问(λ)=λ2 米+ λC+ķ. 它是 Hammarlinget 等人对 quadeig 算法的升级，它试图在 QZ 迭代之前通过通缩来揭示和去除一定数量的零和无限特征值。quadeig 框架的拟议修改旨在增强后向稳定性并使缩小无限和零特征值的过程在数值上更加稳健。特别是，仔细的预处理允许缩放不变/组件方式的后向误差，从而获得更好的条件数。此外，使用 Kronecker 规范形式的上三角版本，除了从米. 对软件实现进行彻底测试的理论分析和经验证据证实了所提出方法的优越数值性能。