江苏省应用数学（中国矿业大学）中心系列学术报告
题目：Solving the quadratic eigenvalue problem expressed in non-monomial basis by the tropically scaled CORK linearization
报告人：汪祥 教授 单位：南昌大学数学与计算机学院
时间：2023年6月13日（周二）上午9:30-10:30
地点：数学院 A302
报告人及报告内容摘要：汪祥，教授、博士生导师。先后入选或获批江西省新世纪百千万人才工程人选，江西省青年科学家，江西省高等学校中青年骨干教师，江西省高水平本科教学团队负责人，江西省优秀研究生指导教师，宝钢全国优秀教师奖获得者；担任中国工业与应用数学学会理事，中国计算数学学会理事，中国高等教育学会数学专委会常务理事, 国家天元数学东南中心执委会委员，国际知名期刊《Computational and Applied Mathematics》的Associate Editor。主要从事数值代数、人工智能与数据科学等领域的研究，在大规模稀疏线性方程组、大规模稀疏特征值问题、线性和非线性矩阵方程的数值求解、谱聚类等方面取得了一些成果。目前主持(含完成)国家自然科学基金3项及省部级项目十几项。近几年以第一作者或通讯作者在国内外权威期刊上共发表SCI收录论文50多篇。以第一完成人身份获江西省自然科学奖三等奖1项和江西省教学成果奖二等奖3项。
Abstract:
In this talk, the quadratic eigenvalue problem (QEP) expressed in various commonly used bases, including Taylor, Newton, and Lagrange basis functions will be introduced. We propose to investigate the backward errors of the computed eigenpairs and condition numbers of eigenvalues incurred by the application of the recently developed and well-received compact rational Krylov (CORK) linearization. To improve the backward error and condition number of QEP expressed in a non-monomial basis, we combine the tropical scaling with the CORK 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 CORK 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 CORK linearization.We unify both bounds and these bounds suggest the tropical scaling to improve the normwise condition number for the CORK linearization and the backward errors of approximate eigenpairs of the QEP obtained from the CORK linearization. Our investigation is accompanied by adequate numerical experiments to justify our theoretical findings.