Advertisement

Numerische Mathematik

, Volume 141, Issue 2, pp 319–351 | Cite as

On the exponential of semi-infinite quasi-Toeplitz matrices

  • Dario A. BiniEmail author
  • Beatrice Meini
Article
  • 46 Downloads

Abstract

Let \(a(z)=\sum _{i\in {\mathbb {Z}}}a_iz^i\) be a complex valued function defined for \(|z|=1\), such that \(\sum _{i\in {\mathbb {Z}}}|a_i|<\infty \); define \(T(a)=(t_{i,j})_{i,j\in {\mathbb {Z}}^+}, t_{i,j}=a_{j-i}\) for \(i,j\in {\mathbb {Z}}^+\), the semi-infinite Toeplitz matrix associated with the symbol a(z); let \(E=(e_{i,j})_{i,j\in {\mathbb {Z}}^+}\) be a compact operator in \(\ell ^p\), with \(1\le p\le \infty .\) A semi-infinite matrix of the kind \(A=T(a)+E\) is said quasi-Toeplitz (QT). The problem of the computation of \(\exp (A)\) or \(\exp (A)v\), with A quasi-Toeplitz and v a vector, arises in many applications. We prove that the exponential of a QT-matrix A is QT, that is, \(\exp (A) = T(\exp (a))+F\) where F is a compact operator in \(\ell ^p\). This property allows the design of an algorithm for computing \(\exp (A)\) and \(\exp (A)v\) up to any precision. The case of families of \(n\times n\) matrices obtained by truncating infinite QT-matrices to finite size is also considered. Numerical experiments show the effectiveness of this approach.

Mathematics Subject Classification

65F60 15A16 15B05 47B35 

Notes

Acknowledgements

The authors wish to thank Robert Luce for providing the software for computing the matrix exponential of a finite Toeplitz matrix based on the displacement rank and the anonymous referees who provided useful suggestions and remarks which helped to improve the presentation of the paper.

References

  1. 1.
    Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011).  https://doi.org/10.1137/100788860 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bini, D., Dendievel, S., Latouche, G., Meini, B.: Computing the exponential of large block-triangular block-Toeplitz matrices encountered in fluid queues. Linear Algebra Appl. 502, 387–419 (2016).  https://doi.org/10.1016/j.laa.2015.03.035 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bini, D.A., Massei, S., Meini, B.: On functions of quasi-Toeplitz matrices. Mat. Sb. 208(11), 56–74 (2017).  https://doi.org/10.4213/sm8864 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bini, D.A., Massei, S., Meini, B.: Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes. Math. Comput. 87(314), 2811–2830 (2018).  https://doi.org/10.1090/mcom/3301 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bini, D.A., Massei, S., Robol, L.: Quasi-Toeplitz matrix arithmetic: a Matlab toolbox. Numer. Algorithms (2018).  https://doi.org/10.1007/s11075-018-0571-6
  6. 6.
    Böttcher, A., Grudsky, S.M.: Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis. Birkhäuser Verlag, Basel (2000).  https://doi.org/10.1007/978-3-0348-8395-5 CrossRefzbMATHGoogle Scholar
  7. 7.
    Böttcher, A., Grusky, S.M.: Spectral Properties of Band Toeplitz Matrices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005)CrossRefGoogle Scholar
  8. 8.
    Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer, Berlin (2012)zbMATHGoogle Scholar
  9. 9.
    Dendievel, S., Latouche, G.: Approximations for time-dependent distributions in Markovian fluid models. Methodol. Comput. Appl. Probab. 19, 285–309 (2016).  https://doi.org/10.1007/s11009-016-9480-0 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fayolle, G., Iasnogorodski, R., Malyshev, V.: Random Walks in the Quarter-Plane. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gavrilyuk, I.P., Hackbusch, W., Khoromskij, B.N.: \({\mathscr {H}}\)-matrix approximation for the operator exponential with applications. Numer. Math. 92(1), 83–111 (2002).  https://doi.org/10.1007/s002110100360 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gavrilyuk, I.P., Makarov, V.L.: Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces. SIAM J. Numer. Anal. 43(5), 2144–2171 (2005).  https://doi.org/10.1137/040611045 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grimm, V.: Resolvent Krylov subspace approximation to operator functions. BIT 52(3), 639–659 (2012).  https://doi.org/10.1007/s10543-011-0367-8 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 1. Wiley, New York (1974)zbMATHGoogle Scholar
  16. 16.
    Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010).  https://doi.org/10.1017/S0962492910000048 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Iserles, A.: How large is the exponential of a banded matrix? Dedicated to John Butcher. New Zealand J. Math. 29(2), 177–192 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kressner, D., Luce, R.: Fast computation of the matrix exponential for a Toeplitz matrix. SIAM J. Matrix Anal. Appl. 39(1), 23–47 (2018).  https://doi.org/10.1137/16M1083633 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley Classics Library. Wiley, New York (1989)zbMATHGoogle Scholar
  21. 21.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    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).  https://doi.org/10.1137/090758064 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Motyer, A.J., Taylor, P.G.: Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators. Adv. Appl. Probab. 38, 522–544 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Corporation, North Chelmsford (1981)zbMATHGoogle Scholar
  25. 25.
    Paige, C.C.: Bidiagonalization of matrices and solutions of the linear equations. SIAM J. Numer. Anal. 11, 197–209 (1974).  https://doi.org/10.1137/0711019 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pang, H.K., Sun, H.W.: Shift-invert Lanczos method for the symmetric positive semidefinite Toeplitz matrix exponential. Numer. Linear Algebra Appl. 18(3), 603–614 (2011).  https://doi.org/10.1002/nla.747 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sakuma, Y., Miyazawa, M.: On the effect of finite buffer truncation in a two-node Jackson network. Stoch. Models 12, 143–164 (2005)zbMATHGoogle Scholar
  28. 28.
    Sericola, B.: Markov Chains. Theory, Algorithms and Applications. Applied Stochastic Methods Series. ISTE, London; Wiley, Hoboken (2013).  https://doi.org/10.1002/9781118731543
  29. 29.
    Shao, M.: On the finite section method for computing exponentials of doubly-infinite skew-Hermitian matrices. Linear Algebra Appl. 451, 65–96 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Takahashi, Y., Fujimoto, K., Makimoto, N.: Geometric decay of the steady-state probabilities in a Quasi-Birth-Death process with a countable number of phases. Stoch. Models 14, 368–391 (2001)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56(3), 385–458 (2014).  https://doi.org/10.1137/130932132 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wu, G., Feng, T.T., Wei, Y.: An inexact shift-and-invert Arnoldi algorithm for Toeplitz matrix exponential. Numer. Linear Algebra Appl. 22(4), 777–792 (2015).  https://doi.org/10.1002/nla.1992 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

Personalised recommendations