Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

  • Dario A. Bini
  • Stefano Massei
  • Leonardo RobolEmail author
Original Paper


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.


Toeplitz matrices Banach algebra MATLAB Wiener algebra Infinite matrices 


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  1. 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. 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. 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. 4.
    Bini, D.A., Böttcher, A: Polynomial factorization through Toeplitz matrix computations. Linear Algebra Appl. 366, 25–37 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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. 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. 7.
    Bini, D.A., Massei, S., Meini, B.: On functions of quasi Toeplitz matrices. Sb. Math. 208(11), 56–74 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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. 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. 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. 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. 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. 13.
    Bini, D.A., Meini, B.: On the exponential of semi-infinite quasi-Toeplitz matrices. arXiv:1611.06380 (2016)
  14. 14.
    Böttcher, A., Grudsky, SM: Spectral properties of banded Toeplitz matrices. SIAM, PA (2005)CrossRefzbMATHGoogle Scholar
  15. 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. 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. 17.
    Böttcher, A., Silbermann, B: Introduction to Large Truncated Toeplitz Matrices. Springer Science & Business Media, Berlin (2012)zbMATHGoogle Scholar
  18. 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. 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. 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. 21.
    Higham, N.J.: Functions of matrices: theory and computation. SIAM, PA (2008)CrossRefzbMATHGoogle Scholar
  22. 22.
    Jackson, J.R.: Networks of waiting lines. Oper. Res. 5(4), 518–521 (1957)MathSciNetCrossRefGoogle Scholar
  23. 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. 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. 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. 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. 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. 28.
    Latouche, G., Taylor, P.: Truncation and augmentation of level-independent QBD processes. Stoch. Process. Appl. 99(1), 53–80 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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. 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. 31.
    Miyazawa, M.: Light tail asymptotics in multidimensional reflecting processes for queueing networks. Top 19(2), 233–299 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Dover Publications, USA (1981)zbMATHGoogle Scholar
  33. 33.
    Paige, C.C.: Bidiagonalization of matrices and solutions of the linear equations. SIAM J. Numer. Anal. 11, 197–209 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants. II Adv. Math. 21(1), 1–29 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPisaItaly
  2. 2.EPF LausanneLausanneSwitzerland
  3. 3.Institute of Information Science and Technologies “A. Faedo”, CNRPisaItaly

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