Journal article  Open Access

Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

Bini D. A., Massei S., Robol L.

Banach algebra  Wiener algebra  Mathematics - Numerical Analysis  FOS: Mathematics  Toeplitz matrices  Infinite matrices  Applied Mathematics  Numerical Analysis (math.NA)  MATLAB 

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 $\lVert A \rVert_{\mathcal QT}=\alpha\lVert a\rVert_{\mathcal{W}} \lVert E \rVert_2$, 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.

Source: Numerical algorithms 81 (2019): 741–769. doi:10.1007/s11075-018-0571-6

Publisher: Baltzer Science Publishers, Bussum ;, Paesi Bassi

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BibTeX entry
	title = {Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox},
	author = {Bini D. A. and Massei S. and Robol L.},
	publisher = {Baltzer Science Publishers, Bussum ;, Paesi Bassi},
	doi = {10.1007/s11075-018-0571-6 and 10.48550/arxiv.1801.08158},
	journal = {Numerical algorithms},
	volume = {81},
	pages = {741–769},
	year = {2019}