The tropical scaling for the polynomial eigenvalue problem solved by a contour integral method
The tropical scaling for the polynomial eigenvalue problem solved by a contour integral method
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轮廓积分法求解多项式特征值问题的热带标度
DOI：
10.1002/nla.2413
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发表时间：
2021
影响因子：
4.3
通讯作者：
Sakurai Tetsuya
中科院分区：
文献类型：
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作者：
Chen Hongjia;Zhang Ke;Sakurai Tetsuya
轮廓积分法是计算多项式特征值问题(PEP)部分特征值的一类有效方法。其中，最近发展起来的基于瑞利-里兹投影的Sakurai-Sugiura方法(SS-RR)因其有效性而备受关注。然而，当投影的PEP的系数矩阵的范数变化很大时，SS-RR方法可能会受到数值不稳定的影响。为了提高数值稳定性，我们将热带尺度技术引入到SS-RR方法中，并建立了原特征值问题的近似特征对向后误差的上界。这些边界揭示了热带尺度提高原SS-RR方法数值稳定性的机制。数值实验表明，该方法能很好地减小实际的后向误差，并能很好地预测定标前后的误差。
The contour integral method is a class of efficient methods for computing partial eigenvalues of polynomial eigenvalue problem (PEP). Among them, the recently developed Sakurai–Sugiura method with Rayleigh–Ritz projection (SS‐RR) method has received much attention for its effectiveness. However, the SS‐RR method may suffer from numerical instability when the coefficient matrices of the projected PEP vary widely in norm. To improve the numerical stability, we incorporate the tropical scaling technique into the SS‐RR method and establish upper bounds for the backward error of an approximate eigenpair of the original eigenvalue problem. These bounds shed light on mechanism that the tropical scaling improves the numerical stability of the original SS‐RR method. Numerical experiments show that the actual backward errors can be successfully reduced by scaling and the bounds can predict well the errors occurring before and after scaling.