计算Toeplitz矩阵的结构化伪谱及其极值点
Computing the structured pseudospectrum of a Toeplitz matrix and its extreme points
AI总结：
本文讨论Toeplitz矩阵关于Frobenius范数的结构化伪谱横坐标和半径的计算，提出两种基于低秩性质的算法构造极值扰动，并扩展至极值点附近的结构化伪谱绘制。
摘要：
讨论了Toeplitz矩阵的结构化伪谱横坐标和半径（关于Frobenius范数）的计算，并提出了两种基于低秩性质构造极值扰动的算法。这些算法受[SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192]中非结构化情况的启发，但将其扩展到结构化伪谱和分析中遇到了若干困难。还讨论了算法的自然推广，允许在极值点附近绘制结构化伪谱的重要部分。由于文献中没有可用于绘制此类结构化伪谱的算法，我们提出的方法似乎有望将现有软件工具（Eigtool, Seigtool）扩展到Toeplitz矩阵的结构化伪谱表示。我们讨论了算法的局部收敛性质，并通过几个示例展示了应用。
英文摘要：
The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing to draw significant sections of the structured pseudospectra in proximity of extremal points are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool, Seigtool) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative examples.
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