二维特征值问题的变分特征和瑞利商迭代及其应用

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-08-06 DOI:10.1137/22m1472589

Tianyi Lu, Yangfeng Su, Zhaojun Bai

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引用次数: 0

摘要

SIAM 矩阵分析与应用期刊》，第 45 卷第 3 期，第 1455-1486 页，2024 年 9 月。摘要。本文介绍了赫米特矩阵对的二维特征值问题（2DEVP）[math]。2DEVP 可视为众所周知的参数矩阵特征值优化问题[math]的线性代数表述。我们首先介绍了 2DEVP 的基本性质，如二维特征值的存在性和变分特征，然后设计了一种类似瑞利商迭代（RQI）的算法，简称 2DRQI，用于计算 2DEVP 的二维特征三元组。2DRQI 的功效通过由瑞利商最小值和稳定矩阵的不稳定性距离引起的大规模特征值优化问题得到了证明。

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Variational Characterization and Rayleigh Quotient Iteration of 2D Eigenvalue Problem with Applications

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1455-1486, September 2024.
Abstract. A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair [math] is introduced in this paper. The 2DEVP can be regarded as a linear algebra formulation of the well-known eigenvalue optimization problem of the parameter matrix [math]. We first present fundamental properties of the 2DEVP, such as the existence and variational characterizations of 2D-eigenvalues, and then devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The efficacy of the 2DRQI is demonstrated by large scale eigenvalue optimization problems arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix.

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.