江蘇省應用數學（中國礦業大學）中心系列學術報告
題目：Solving the quadratic eigenvalue problem expressed in non-monomial basis by the tropically scaled CORK linearization
報告人：汪祥 教授 單位：南昌大學數學與計算機學院
時間：2023年6月13日（周二）上午9:30-10:30
地點：數學院 A302
報告人及報告内容摘要：汪祥，教授、博士生導師。先後入選或獲批江西省新世紀百千萬人才工程人選，江西省青年科學家，江西省高等學校中青年骨幹教師，江西省高水平本科教學團隊負責人，江西省優秀研究生指導教師，寶鋼全國優秀教師獎獲得者；擔任中國工業與應用數學學會理事，中國計算數學學會理事，中國高等教育學會數學專委會常務理事, 國家天元數學東南中心執委會委員，國際知名期刊《Computational and Applied Mathematics》的Associate Editor。主要從事數值代數、人工智能與數據科學等領域的研究，在大規模稀疏線性方程組、大規模稀疏特征值問題、線性和非線性矩陣方程的數值求解、譜聚類等方面取得了一些成果。目前主持(含完成)國家自然科學基金3項及省部級項目十幾項。近幾年以第一作者或通訊作者在國内外權威期刊上共發表SCI收錄論文50多篇。以第一完成人身份獲江西省自然科學獎三等獎1項和江西省教學成果獎二等獎3項。
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.