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Backward error analysis of linearizing–balancing strategies for heavily damped quadratic eigenvalue problem
Applied Mathematics Letters ( IF 3.7 ) Pub Date : 2021-04-24 , DOI: 10.1016/j.aml.2021.107342
Hongjia Chen , Lei Du , Zongqi Cao

A classical approach for solving quadratic eigenvalue problem (QEP) is via linearization. However, it can suffer from numerical instability when the norms of coefficient matrices vary widely. Two strategies of Betcke’s balancing for heavily damped QEP are considered. One strategy is to balance matrix polynomial before linearizing (called balancing–linearizing for short). The other strategy is first to linearize the matrix polynomial, then balancing (called linearizing–balancing for short). We analyze the backward error of approximate eigenpairs computed by these two strategies, and find that the backward error of approximate eigenpairs by linearizing–balancing is smaller than that by balancing–linearizing under relatively mild conditions. Numerical experiments are presented to demonstrate the advantages of linearizing-balancing.



中文翻译:

重阻尼二次特征值问题线性均衡策略的后向误差分析

解决二次特征值问题(QEP)的经典方法是通过线性化。但是,当系数矩阵的范数变化很大时,它可能会遭受数值不稳定性的困扰。考虑了Betcke针对高阻尼QEP进行平衡的两种策略。一种策略是在线性化之前平衡矩阵多项式(简称平衡-线性化)。另一种策略是首先线性化矩阵多项式,然后进行平衡(简称线性化-平衡)。我们分析了这两种策略计算出的近似本征对的后向误差,发现在相对温和的条件下,通过线性化-平衡,近似本征对的后向误差小于通过平衡-线性化。数值实验表明了线性平衡的优势。

更新日期:2021-04-30
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