准Toeplitz矩阵运算：一个MATLAB工具箱
Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox
AI总结：
本文提出了一种用于准Toeplitz矩阵运算的MATLAB工具箱，通过有限参数实现准Toeplitz矩阵的运算，并扩展到有限大小的Toeplitz加低秩结构矩阵，从而高效解决相关矩阵函数和方程问题。
摘要：
准Toeplitz（QT）矩阵是一种半无限矩阵，形式为A=T(a)+E，其中T(a)=(a_{j-i})_{i,j∈Z^+}，E=(e_{i,j})_{i,j∈Z^+}为紧致矩阵，且范数‖a‖_W=∑_{i∈Z}|a_i|和‖E‖_2均为有限。这些性质允许通过有限参数近似任何QT矩阵。QT矩阵配备范数‖A‖_QT=α‖a‖_W‖E‖_2（α=(1+√5)/2）构成Banach代数。本文描述了这些运算的有限参数算法，并开发了一个MATLAB工具箱实现这些运算。该工具箱进一步扩展到有限大小的Toeplitz加低秩结构矩阵，从而开发出成本不随问题维度增加的算法。一些应用于矩阵函数计算和矩阵方程求解的例子展示了该方法的有效性。
英文摘要：
A Quasi Toeplitz (QT) matrix is a semi-infinite matrix of the kind $A=T(a)+E$ where $T(a)=(a_{j-i})_{i,j\in\mathbb Z^+}$, $E=(e_{i,j})_{i,j\in\mathbb Z^+}$ is compact and the norms $\lVert a\rVert_{\mathcal W} = \sum_{i\in\mathbb Z}|a_i|$ and $\lVert E \rVert_2$ are finite. These properties allow to approximate any QT-matrix, within any given precision, by means of a finite number of parameters. QT-matrices, equipped with the norm $\lVert A \rVert_{\mathcal QT}=α\lVert a\rVert_{\mathcal{W}} \lVert E \rVert_2$, for $α= (1+\sqrt 5)/2$, are a Banach algebra with the standard arithmetic operations. We provide an algorithmic description of these operations on the finite parametrization of QT-matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
