基于预解式有理逼近的非线性特征值求解器的收敛性分析
Convergence analysis of a nonlinear eigensolver based on rational approximation of the resolvent
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中文总结 AI 辅助
研究基于预解式有理逼近的非线性特征值求解器，分析其收敛性，证明块探测和放大两种提高精度技术的有效性，通过广义特征问题建立极点查找稳定性，数值实验验证理论结果。
中文摘要 AI 辅助
给定一个全纯矩阵值函数，其草图预解式的极点通常是其特征值。一旦获得预解式的良好有理逼近，该有理逼近的极点通常接近那些特征值，为求解线性和非线性特征值问题提供了灵活框架。但计算特征值的精度有限且了解不足。本文分析了该方法的收敛性，并证明了两种提高精度的技术（块探测和放大）的有效性。还通过广义特征问题建立了重心有理形式极点查找的前后稳定性。数值实验证明了理论结果的精确性。
英文摘要
Given a holomorphic matrix-valued function, the poles of its sketched resolvent are generically its eigenvalues. Once a good rational approximation of the sketched resolvent is obtained, the poles of this rational approximation typically lie close to those eigenvalues, thus providing a flexible framework for solving both linear and nonlinear eigenvalue problems. However, the accuracy of the computed eigenvalues is limited and remains poorly understood. This paper analyzes the convergence of this approach and demonstrates the effectiveness of two techniques to improve accuracy: block probing and zooming in. We also establish the backward and forward stability of polefinding for a barycentric rational form via a generalized eigenproblem. Numerical experiments demonstrate the sharpness of our theoretical results.
