报告题目：ON THE *-CONGRUENCE SYLVESTER EQUATION
报告人：Shao-Liang Zhang, Professor of Computational Mathematics at Department of Applied Physics, Graduate School of Engineering, Nagoya University, Japan, (2017—Present)
报告时间：2024年9月27日 10:00-11:00
报告地点：数学学院三楼研讨室5
校内联系人：邹永魁 zouyk@jlu.edu.cn
报告摘要：We consider the following matrix equation
AX + X*B = C, (1)
where A ∈ C^(m×n), B ∈ C^(n×m), and C ∈ C^(m×m) are given, and X ∈ C^(n×m) is to be determined. The operator (·)* denotes the transpose (·)T or the conjugate transpose (·)H of a matrix. Equation (1) is called the *-congruence Sylvester equation. The *-congruence Sylvester equation appears in palindromic eigenvalue problems arising from some realistic applications such that the vibration analysis of fast trains, see, e.g., [1].
In recent paper [2], when *= T and the given matrices are square (m = n), it was shown that the *-congruence Sylvester equation is mathematically equivalent to the Lyapunov equation under certain conditions. The Lyapunov equation is widely known in control theory and has been much studied by many researchers. Therefore, the study [2] indicates that it can be possible to utilize the rich literature on the Lyapunov equation for the ⋆-congruence Sylvester equation (1). Indeed, it was shown that the direct method for the Lyapunov equation can be used to obtain a numerical solution of (1) with lower computational cost than the conventional method [4]. One of the important approaches in the previous study is to use an equivalent linear system that is obtained by vectorization. Oozawa et al. succeeded in reducing (1) to the Lyapunov equation by applying an appropriate linear operator to the linear system and returning it into a matrix.
The theoretical result was extended to the case where A and B are rectangular (m is not equal to n). In this case, it was shown that (1) is equivalent to the generalized Sylvester equation [3]. However, when * = H, the same transformation cannot be applied to (1) because (1) is a nonlinear equation.
In this talk, we consider a linearization of (1) with * = H, and construct an appropriate operator to obtain an equivalent linear matrix equation. We show that (1) is equivalent to the generalized Sylvester equation under certain conditions, and introduce numerical solvers utilizing our results.
References
[1] D. Kressner et al., Numer. Algorithms, 51(2): 209–238, 2009.
[2] M. Oozawa et al., J. Comput. Appl. Math., 329: 51–56, 2018.
[3] Y. Satake et al., Appl. Math. Lett., 96: 7–13, 2019.
[4] F. D. Terán and F. M. Dopico, Electron. J. Linear Algebra, 22: 849–863, 2011.
报告人简介：张绍良教授于1983年在吉林大学获得学士学位(计算数学)，于1990年在日本Tsukuba大学获得博士学位。
Academic Experiences
• Professor of Computational Mathematics at Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Japan(2005—2017)
• Professor of Computational Mathematics at Department of Applied Physics, Graduate School of Engineering, University of Tokyo, Japan(2005—2007)
• Associate Professor of Computational Mathematics at Department of Applied Physics, Graduate School of Engineering, University of Tokyo, Japan(1998—2005)
• Lecturer of Computational Mathematics at Institute of Information Sciences and Electronics, University of Tsukuba, Japan(1995—1998)
• Assistant Professor of Computational Mathematics at Department of Information Engineering, Faculty of Engineering, Nagoya University, Japan(1993—1995)
• Research Scientist of Computational Mathematics at Institute of Computational Fluid Dynamics, Japan(1990—1993)
Other Professional Activities
• Secretary of East Asia Section of the Society for Industrial and Applied Mathematics(2011— 2012)
• Director of the China Center for International Exchange, Nagoya University(2014—Present)
• Vice President of Japan Society for Industrial and Applied Mathematics(2021—2023)
• Fellow of Japan Society for Industrial and Applied Mathematics(2022—Present)
Awards
• Best Paper Awards of Japan Society for Industrial and Applied Mathematics, 1997
• Best Paper Awards of Japan Society for Industrial and Applied Mathematics, 2002
• Best Paper Awards of Japan Society for Industrial and Applied Mathematics, 2010
• Best Paper Awards of Japan Society for Industrial and Applied Mathematics, 2011
