# Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

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## 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.

## Keywords

Toeplitz matrices Banach algebra MATLAB Wiener algebra Infinite matrices## Preview

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## References

- 1.Bean, N., Latouche, G.: Approximations to quasi-birth-and-death processes with infinite blocks. Adv. Appl. Probab.
**42**(4), 1102–1125 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Bini, D., Massei, S., Meini, B., Robol, L.: On Quadratic Matrix Equations with Infinite Size Coefficients Encountered in QBD Stochastic Processes. Numerical Linear Algebra Application, in pressGoogle Scholar
- 3.Bini, D., Pan, V.Y: Polynomial and Matrix Computations, vol. 1. Progress in Theoretical Computer Science. Birkhäuser Boston, Inc, Boston (1994). Fundamental algorithmsCrossRefzbMATHGoogle Scholar
- 4.Bini, D.A., Böttcher, A: Polynomial factorization through Toeplitz matrix computations. Linear Algebra Appl.
**366**, 25–37 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Bini, D.A., Fiorentino, G., Gemignani, L., Meini, B.: Effective fast algorithms for polynomial spectral factorization. Numer. Algorithm.
**34**(2-4), 217–227 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Bini, D.A., Gemignani, L., Meini, B.: Computations with infinite Toeplitz matrices and polynomials. Linear Algebra Appl.
**343**(/344), 21–61 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Bini, D.A., Massei, S., Meini, B.: On functions of quasi Toeplitz matrices. Sb. Math.
**208**(11), 56–74 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Bini, D.A., Massei, S., Meini, B.: Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes. Math. Comp. (2018)Google Scholar
- 9.Bini, D.A., Massei, S., Robol, L.: Efficient cyclic reduction for quasi-birth-death problems with rank structured blocks. Appl. Numer Math.
**116**, 37–46 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Bini, D.A., Massei, S., Robol, L.: On the decay of the off-diagonal singular values in cyclic reduction. Linear Algebra Appl.
**519**, 27–53 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Bini, D.A., Meini, B.: Effective methods for solving banded Toeplitz systems. SIAM J. Matrix Anal. Appl.
**20**(3), 700–719 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Bini, D.A., Meini, B.: The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond. Numer. Algorithm.
**51**(1), 23–60 (2009)CrossRefzbMATHGoogle Scholar - 13.Bini, D.A., Meini, B.: On the exponential of semi-infinite quasi-Toeplitz matrices. arXiv:1611.06380 (2016)
- 14.Böttcher, A., Grudsky, SM: Spectral properties of banded Toeplitz matrices. SIAM, PA (2005)CrossRefzbMATHGoogle Scholar
- 15.Böttcher, A., Halwass, M.: A Newton method for canonical Wiener-Hopf and spectral factorization of matrix polynomials. Electron. J. Linear Algebra
**26**, 873–897 (2013)MathSciNetzbMATHGoogle Scholar - 16.Böttcher, A., Halwass, M.: Wiener-Hopf and spectral factorization of real polynomials by Newton’s method. Linear Algebra Appl.
**438**(12), 4760–4805 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Böttcher, A., Silbermann, B: Introduction to Large Truncated Toeplitz Matrices. Springer Science & Business Media, Berlin (2012)zbMATHGoogle Scholar
- 18.Gohberg, I.C.: On an application of the theory of normed rings to singular integral equations. Uspehi Matem. Nauk. (N.S.)
**7**(2(48)), 149–156 (1952)MathSciNetGoogle Scholar - 19.Gutiérrez-Gutiérrez, J, Crespo, P.M., Böttcher, A: Functions of the banded Hermitian block Toeplitz matrices in signal processing. Linear Algebra Appl.
**422**(2-3), 788–807 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Halko, N., Martinsson, P.-G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev.
**53**(2), 217–288 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Higham, N.J.: Functions of matrices: theory and computation. SIAM, PA (2008)CrossRefzbMATHGoogle Scholar
- 22.Jackson, J.R.: Networks of waiting lines. Oper. Res.
**5**(4), 518–521 (1957)MathSciNetCrossRefGoogle Scholar - 23.Kapodistria, S., Palmowski, Z.: Matrix geometric approach for random walks. Stability condition and equilibrium distribution. Stoch. Model.
**33**(4), 572–597 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Kobayashi, M., Miyazawa, M.: Revisiting the tail asymptotics of the double QBD process: refinement and complete solutions for the coordinate and diagonal directions. In: Matrix-Analytic Methods in Stochastic Models, Volume 27 of Springer Proc. Math. Stat., pp. 145–185. Springer, New York (2013)Google Scholar
- 25.Kressner, D., Luce, R.: Fast computation of the matrix exponential for a Toeplitz matrix. SIAM J. Matrix Anal. Appl.
**39**(1), 23–47 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Latouche, G., Nguyen, G.T., Taylor, P.G.: Queues with boundary assistance: the effects of truncation. Queueing Syst.
**69**(2), 175–197 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM Philadelphia, PA (1999)CrossRefzbMATHGoogle Scholar
- 28.Latouche, G., Taylor, P.: Truncation and augmentation of level-independent QBD processes. Stoch. Process. Appl.
**99**(1), 53–80 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 29.Lee, S.T., Pang, H.-K., Sun, H.-W.: Shift-invert ARnoldi approximation to the Toeplitz matrix exponential. SIAM J. Sci. Comput.
**32**(2), 774–792 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 30.Lindner, M.: Infinite Matrices and Their Finite Sections. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). An introduction to the limit operator methodGoogle Scholar
- 31.Miyazawa, M.: Light tail asymptotics in multidimensional reflecting processes for queueing networks. Top
**19**(2), 233–299 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Dover Publications, USA (1981)zbMATHGoogle Scholar
- 33.Paige, C.C.: Bidiagonalization of matrices and solutions of the linear equations. SIAM J. Numer. Anal.
**11**, 197–209 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 34.Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants. II Adv. Math.
**21**(1), 1–29 (1976)MathSciNetCrossRefzbMATHGoogle Scholar