一、主讲人介绍：Dario A. Bini

Dario A. Bini，意大利比萨大学数学院教授，主要从事马尔可夫链及排队问题数值解、矩阵方程数值解法、结构化矩阵计算、几何矩阵均值及其算法。Dario A. Bini教授在Numerische Mathematik，Mathematics of Computation，SIAM Journal on Scientific Computing，SIAM Journal on Matrix Analysis and Applications，IMA Journal of Numerical Analysis，Numerical Linear Algebra with Applications等计算数学国际顶尖和权威期刊发表论文200余篇，出版Numerical Solution of Algebraic Riccati Equations，Numerical Methods for Structured Markov Chains等计算数学论著7余篇。曾担任SIAM J. Matrix Analysis Appl.，Electronic Transactions on Numerical Analysis，Electronic Journal of Linear Algebra等计算数学国际顶尖以及权威期刊编委。

二、讲座信息

讲座摘要：

We consider the problem of solving matrix equations of the kind A_1 X^2+A_0X+A_(-1)=X , where the coefficients  A_r ,r=-1,0,1, are matrices having specific structures, and X is the unknown matrix. The solution of interest is the one that has some minimality properties, say, it has a minimal spectral radius or has nonnegative entries with minimal value. This kind of problem is encountered in the solution of Quasi-Birth-Death processes, a general framework that models real-world problems in terms of Markov chains. In this talk, after presenting and motivating the interest of this class of equations, we investigate some computational issues encountered in their solution. For this class of problems, the coefficients A_r ,r=-1,0,1 ,  are semi-infinite Quasi-Toeplitz (QT) matrices. We give conditions under which the class of QT matrices is a Banach algebra, that is, a vector space closed under multiplication, endowed with a norm that makes it a Banach space. We give conditions under which the sought solution, say the minimal nonnegative one, is still a QT matrix, and describe and analyze  algorithms for its effective computation. Finally, by means of some numerical experiments performed with the CQT Matlab Toolbox, we show the effectiveness of our algorithms。

讲座时间：5月26日（星期四）13:30-14:30

腾讯会议号：142-518-059

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数学科学学院  

2022年5月18日