Speaker: Professor Wang Xiang
Topic: Solving the quadratic eigenvalue problem expressed in non-monomial basis by the tropically scaled CORK linearization
Date: June 13, 2023
Time: 3:30 - 4:30 p.m.
Venue: Academic Lecture Hall 204, Building 9
Sponsors: School of Mathematics and Statistics, Research Institute of Mathematical Science, Institute of Science and Technology
Biography：
Prof. Wang Xiang is a doctoral supervisor. He has been selected as a young scientist, a young and middle-aged backbone teacher in higher education institutions, the leader of a high-level undergraduate teaching team and an excellent supervisor of graduate students in Jiangxi Province. He also received the Baosteel National Excellent Teacher Award. He serves as director of the China Society for Industrial and Applied Mathematics, director of the Chinese Society of Computational Mathematics, standing director of the Mathematics Special Committee of the China Association of Higher Education, and a member of the Executive Committee of the Tianyuan Mathematical Center in Southeast China. He is also an Associate Editor of the internationally renowned journal Computational and Applied Mathematics.
Professor Wang mainly engages in research on numerical algebra, artificial intelligence and data science. He has made significant achievements in areas such as large-scale sparse linear systems, large-scale sparse eigenvalue problems, numerical solutions for linear and nonlinear matrix equations, and spectral clustering. He is leading or has completed three projects funded by the National Natural Science Foundation and more than ten provincial-level projects. In recent years, he has published over 50 SCI-indexed papers as the first author or corresponding author in domestic and international journals. He has also been awarded the third prize of the Jiangxi Provincial Natural Science Award and 3 second prizes of the Jiangxi Provincial Teaching Achievement Award as the leader of the projects.
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.