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Quasi-Toeplitz matrix arithmetic: a MATLAB toolbox

  • Dario A. Bini
  • Stefano Massei
  • Leonardo Robol
Original Paper
  • 22 Downloads

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 

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References

  1. 1.
    Bean, N., Latouche, G.: Approximations to quasi-birth-and-death processes with infinite blocks. Adv. Appl. Probab. 42(4), 1102–1125 (2010)MathSciNetCrossRefMATHGoogle 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 algorithmsCrossRefMATHGoogle Scholar
  4. 4.
    Bini, D.A., Böttcher, A: Polynomial factorization through Toeplitz matrix computations. Linear Algebra Appl. 366, 25–37 (2003)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bini, D.A., Massei, S., Meini, B.: On functions of quasi Toeplitz matrices. Sb. Math. 208(11), 56–74 (2017)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Böttcher, A., Silbermann, B: Introduction to Large Truncated Toeplitz Matrices. Springer Science & Business Media, Berlin (2012)MATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Higham, N.J.: Functions of matrices: theory and computation. SIAM, PA (2008)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  28. 28.
    Latouche, G., Taylor, P.: Truncation and augmentation of level-independent QBD processes. Stoch. Process. Appl. 99(1), 53–80 (2002)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Dover Publications, USA (1981)MATHGoogle Scholar
  33. 33.
    Paige, C.C.: Bidiagonalization of matrices and solutions of the linear equations. SIAM J. Numer. Anal. 11, 197–209 (1974)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants. II Adv. Math. 21(1), 1–29 (1976)MathSciNetCrossRefMATHGoogle 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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