本文研究了半无限准Toeplitz矩阵的性质，并提出了一种有限表示方法和相关算法，用于解决涉及可数相态的Quasi-Birth and Death (QBD)随机过程的二次矩阵方程。
摘要：
令$\mathcal{W}_1$表示形如$a(z)=\sum_{i=-\infty}^{+\infty}a_iz^i$的复值函数集合，这些函数在单位圆上连续，且满足$\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$。我们称CQT矩阵为与连续符号$a(z)\in\mathcal W_1$相关的准Toeplitz矩阵$A$，形式为$A=T(a)+E$，其中$T(a)=(t_{i,j})_{i,j\in\mathbb{Z}^+}$是半无限Toeplitz矩阵，使得$t_{i,j}=a_{j-i}$，对于$i,j\in\mathbb Z^+$，而$E=(e_{i,j})_{i,j\in\mathbb{Z}^+}$是半无限矩阵，使得$\sum_{i,j=1}^{+\infty}|e_{i,j}|$是有限的。我们证明CQT矩阵的类别在合适的子乘性矩阵范数$\|\cdot\|$下是一个Banach代数。我们引入了CQT矩阵的有限表示方法，并提出了实现基本矩阵运算的算法。一个应用是解决二次矩阵方程$AX^2+BX+C=0$，这在求解具有可数相态的Quasi-Birth and Death (QBD)随机过程时遇到，其中$A,B,C$都是CQT矩阵。
英文摘要：
Denote by $\mathcal{W}_1$ the set of complex valued functions of the form $a(z)=\sum_{i=-\infty}^{+\infty}a_iz^i$ which are continuous on the unit circle, and such that $\sum_{i=-\infty}^{+\infty}|ia_i|<\infty$. We call CQT matrix a quasi-Toeplitz matrix $A$, associated with a continuous symbol $a(z)\in\mathcal W_1$, of the form $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\mathbb{Z}^+}$ is the semi-infinite Toeplitz matrix such that $t_{i,j}=a_{j-i}$, for $i,j\in\mathbb Z^+$, and $E=(e_{i,j})_{i,j\in\mathbb{Z}^+}$ is a semi-infinite matrix such that $\sum_{i,j=1}^{+\infty}|e_{i,j}|$ is finite. We prove that the class of CQT matrices is a Banach algebra with a suitable sub-multiplicative matrix norm $\|\cdot\|$. We introduce a finite representation of CQT matrices together with algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind $AX^2+BX+C=0$, encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented where $A,B,C$ are CQT matrices.
Condition numbers of block Toeplitz matrices and stability of space-time IgA approximations for the wave and Schrödinger equations
Universidad del Valle（瓦列大学）；Technische Universität Chemnitz（开姆尼茨工业大学）；Università di Pavia（帕维亚大学）；CINVESTAV del IPN（墨西哥国立理工学院高级研究中心）；Southern Federal University（南方联邦大学）；Università degli Studi dell’Insubria（因苏布里亚大学）；Uppsala University（乌普萨拉大学）
