Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

Abstract

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 $\|a\|_{_{\mathcal {W}}}={\sum }_{i\in \mathbb Z}|a_{i}|$ and $\|E\|_{2}$ 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 $\|A\|_{_{\mathcal {Q}\mathcal {T}}}=\alpha {\|a\|}_{_{\mathcal {W}}}+\|E\|_{2}$ , for $\alpha = (1+\sqrt 5)/2$ , 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.