Solving the quadratic eigenvalue problem expressed in non-monomial bases by the tropical scaling

IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED
Hongjia Chen, Teng Wang, Chun-Hua Zhang, Xiang Wang
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引用次数: 0

Abstract

In this paper, we consider the quadratic eigenvalue problem (QEP) expressed in various commonly used bases, including Taylor, Newton, and Lagrange bases. We propose to investigate the backward errors of the computed eigenpairs and condition numbers of eigenvalues for QEP solved by a class of block Kronecker linearizations. To improve the backward error and condition number of the QEP expressed in a non-monomial basis, we combine the tropical scaling with the block Kronecker linearization. We then establish upper bounds for the backward error of an approximate eigenpair of the QEP relative to the backward error of an approximate eigenpair of the block Kronecker linearization with and without tropical scaling. Moreover, we get bounds for the normwise condition number of an eigenvalue of the QEP relative to that of the block Kronecker linearization. Our investigation is accompanied by adequate numerical experiments to justify our theoretical findings.

用热带标度法求解非一元基表示的二次特征值问题
本文考虑了用泰勒基、牛顿基和拉格朗日基等几种常用基表示的二次特征值问题。我们提出研究用一类块Kronecker线性化方法求解QEP的计算特征对和特征值的条件数的后向误差。为了改善非一元基表示的QEP的后向误差和条件数,我们将热带标度与块Kronecker线性化相结合。然后,我们建立了QEP的近似特征对相对于具有和不具有热带尺度的块Kronecker线性化的近似特征对的向后误差的上界。此外,我们得到了相对于块Kronecker线性化的QEP特征值的正态条件数的界。我们的研究伴随着足够的数值实验来证明我们的理论发现。
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来源期刊
CiteScore
3.00
自引率
5.90%
发文量
68
审稿时长
3 months
期刊介绍: Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis. This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.
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