A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the kind A= T(a) + E where T(a)=(aj−i)i,j∈ℤ+, E=(ei,j)i,j∈ℤ+ is compact and the norms∥a∥W=∑i∈ℤ|ai| 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∥QT=α∥a∥W+∥E∥2, for α=(1+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.

Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

Bini D. A.;Massei S.;Robol L.
2019

Abstract

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the kind A= T(a) + E where T(a)=(aj−i)i,j∈ℤ+, E=(ei,j)i,j∈ℤ+ is compact and the norms∥a∥W=∑i∈ℤ|ai| 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∥QT=α∥a∥W+∥E∥2, for α=(1+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.
Bini, D. A.; Massei, S.; Robol, L.
File in questo prodotto:
File Dimensione Formato  
cqt-toolbox-paper.pdf

accesso aperto

Descrizione: Post-print
Tipologia: Documento in Post-print
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 490.83 kB
Formato Adobe PDF
490.83 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/1000235
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 11
  • ???jsp.display-item.citation.isi??? 11
social impact