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Functions of rational Krylov space matrices and their decay properties

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Abstract

Rational Krylov subspaces have become a fundamental ingredient in numerical linear algebra methods associated with reduction strategies. Nonetheless, many structural properties of the reduced matrices in these subspaces are not fully understood. We advance in this analysis by deriving bounds on the entries of rational Krylov reduced matrices and of their functions, that ensure an a-priori decay of their entries as we move away from the main diagonal. As opposed to other decay pattern results in the literature, these properties hold in spite of the lack of any banded structure in the considered matrices. Numerical experiments illustrate the quality of our results.

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Notes

  1. Note that Dzhrbashyan is often also transliterated as Djrbashian. We chose to use Dzhrbashyan in accordance with [44].

  2. As remarked in [3], the bound \(\Vert {\mathcal {F}}_{+} \Vert \le 2\) was implicitly given in [30]; moreover, the same bound stands for the operator \({\mathcal {F}}_{-}(g)(w) := {\mathcal {F}}(g)(w) - g(0)\) and can be obtained, e.g., modifying the proof of Theorem 2 in [17, p. 49].

  3. Note that in Sect. 7.1 we proved that Proposition 2.1 holds also for \({\widetilde{J}}_m\).

  4. The index \(j=3,4,\dots \) parametrizes the family of sequences of FD rational functions \(\{M_t^{(j)}\}_{t=0,\,\dots }\). All sequences in such family share the initial (finite) shifts \(\phi (\sigma _{s+1})\) up to \(j-3\). This means that given the indexes \(i<j\), it holds that \(M_t^{(i)} = M_t^{(j)}\) for \(t=0,\dots ,i-3\).

References

  1. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems, Vol.6 of Advances in Design and Control. SIAM, Philadelphia, PA (2005). With a foreword by Jan C. Willems

    Book  Google Scholar 

  2. Beckermann, B.: An error analysis for rational Galerkin projection applied to the Sylvester equation. SIAM J. Numer. Anal. 49, 2430–2450 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beckermann, B., Reichel, L.: Error estimates and evaluation of matrix functions via the Faber transform. SIAM J. Numer. Anal. 47, 3849–3883 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.): Model Reduction and Approximation: Theory and Algorithms. Computational Science & Engineering. SIAM, Philadelphia, PA (2017)

    Google Scholar 

  5. Benner, P., Mehrmann, V., Sorensen, D. (eds.): Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2005)

    Google Scholar 

  6. Benzi, M., Razouk, N.: Decay bounds and \(O(n)\) algorithms for approximating functions of sparse matrices. ETNA 28, 16–39 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Benzi, M., Simoncini, V.: Decay bounds for functions of Hermitian matrices with banded or Kronecker structure. SIAM J. Matrix Anal. Appl. 36, 1263–1282 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bouras, A., Frayssé, V.: Inexact matrix-vector products in Krylov methods for solving linear systems: a relaxation strategy. SIAM J. Matrix Anal. Appl. 26, 660–678 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Deckers, K., Bultheel, A.: Rational Krylov sequences and orthogonal rational functions. Technical report, Department of Computer Science, K.U.Leuven (2007)

  10. Demko, S., Moss, W.F., Smith, P.W.: Decay rates for inverses of band matrices. Math. Comput. 43, 491–499 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Druskin, V., Knizhnerman, L.: Extended Krylov subspaces: approximation of the matrix square root and related functions. SIAM J. Matrix Anal. Appl. 19, 755–771 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Druskin, V., Knizhnerman, L., Simoncini, V.: Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation. SIAM J. Numer. Anal. 49, 1875–1898 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Druskin, V., Knizhnerman, L., Zaslavsky, M.: Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with optimized shifts. SIAM J. Sci. Comput. 31, 3760–3780 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Druskin, V., Simoncini, V.: Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst. Control Lett. 60, 546–560 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dzhrbashyan, M.M.: On expansion of analytic functions in rational functions with preassigned poles. Izv. Akad. Nauk Armyan. SSR. Ser. Fiz.-Mat. Nauk 10, 21–29 (1957)

    MathSciNet  Google Scholar 

  16. Dzhrbashyan, M.M.: Expansions in rational functions with fixed poles. Dokl. Akad. Nauk SSSR 143, 17–20 (1962)

    MathSciNet  MATH  Google Scholar 

  17. Gaier, D.: Lectures on Complex Approximation. Birkhäuser Boston, Inc., Boston, MA (1987). Translated from the German by Renate McLaughlin

    Book  MATH  Google Scholar 

  18. Gaier, D.: The Faber operator and its boundedness. J. Approx. Theory 101, 265–277 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ganelius, T.: Degree of rational approximation. In: Lectures on Approximation and Value Distribution, vol. 79 of Sém. Math. Sup., Presses Univ. Montréal, Montreal, Que., pp. 9–78 (1982)

  20. Güttel, S.: Rational Krylov methods for operator functions. Ph.D. thesis, TU Bergakademie Freiberg, Germany (2010)

  21. Güttel, S.: Rational Krylov approximation of matrix functions: numerical methods and optimal pole selection. GAMM-Mitteilungen 36, 8–31 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Güttel, S., Knizhnerman, L.: A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions. BIT Numer. Math. 53, 595–616 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Güttel, S., Schweitzer, M.: A comparison of limited-memory Krylov methods for Stieltjes functions of Hermitian matrices. SIAM J. Matrix. Anal. Appl. 42(1):83–107 (2021)

  24. Higham, N.J.: Functions of Matrices. Theory and Computation. SIAM, Philadelphia, PA (2008)

    Book  MATH  Google Scholar 

  25. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  26. Jagels, C., Reichel, L.: The extended Krylov subspace method and orthogonal Laurent polynomials. Linear Algebra Appl. 431, 441–458 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jagels, C., Reichel, L.: The structure of matrices in rational Gauss quadrature. Math. Comput. 82, 2035–2060 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Knizhnerman, L., Simoncini, V.: Convergence analysis of the extended Krylov subspace method for the Lyapunov equation. Numer. Math. 118, 567–586 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Korvink, J.G., Rudnyi, E.B.: Oberwolfach benchmark collection. In: Benner, P., Sorensen, D.C., Mehrmann, V. (eds.) Dimension Reduction of Large-Scale Systems, pp. 311–315. Springer, Berlin (2005)

    Chapter  Google Scholar 

  30. Kövari, T., Pommerenke, C.: On Faber polynomials and Faber expansions. Math. Z. 99, 193–206 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kürschner, P., Freitag, M.: Inexact methods for the low rank solution to large scale Lyapunov equations. BIT Numer. Math. 60, 1221–1259 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lancaster, P.: Explicit solutions of linear matrix equations. SIAM Rev. 12, 544–566 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  33. Malmquist, F.: Sur la détermination d’une classe de fonctions analytiques par leurs valeurs dans un ensemble donné de points, Comptes Rendus du Sixièmme Congrèss (1925) des mathématiciens scandinaves. Kopenhagen, pp. 253–259 (1926)

  34. Nabben, R.: Decay rates of the inverse of nonsymmetric tridiagonal and band matrices. SIAM J. Matrix Anal. Appl. 20, 820–837 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Olsson, K.H.A., Ruhe, A.: Rational Krylov for eigenvalue computation and model order reduction. BIT Numer. Math. 46, 99–111 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pozza, S., Simoncini, V.: Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices. BIT Numer. Math. 59, 969–986 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ruhe, A.: Rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl. 58, 391–405 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ruhe, A.: The rational Krylov algorithm for nonsymmetric eigenvalue problems III Complex shifts for real matrices. BIT Numer. Math. 34, 165–176 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Simoncini, V.: Variable accuracy of matrix-vector products in projection methods for eigencomputation. SIAM J. Numer. Anal. 43, 1155–1174 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Simoncini, V.: The extended Krylov subspace for parameter dependent systems. Appl. Numer. Math. 60, 550–560 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Simoncini, V.: The Lyapunov matrix equation. Matrix analysis from a computational perspective. In: Quaderno UMI—Topics in Mathematics, UMI, vol. 55, pp. 157–174 (2015)

  42. Simoncini, V.: Computational methods for linear matrix equations. SIAM Rev. 58, 377–441 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Simoncini, V., Szyld, D.B.: Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM J. Sci. Comput. 25, 454–477 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  44. Suetin, P.K.: Series of Faber Polynomials. Gordon and Breach Science Publishers, London (1998). Translated from the 1984 Russian original by E. V. Pankratiev [E. V. Pankrat\(^{\prime }\)ev]

    MATH  Google Scholar 

  45. Takenaka, S.: On the orthogonal functions and a new formula of interpolation. Jpn. J. Math. 2, 129–145 (1925)

    Article  MATH  Google Scholar 

  46. Walsh, J.L.: Interpolation and Approximation by Rational Functions in the Complex Domain, vol. XX, 4th edn. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI (1965)

    MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for the helpful comments and improvements suggested. The authors are members of Indam-GNCS, which support is gratefully acknowledged. This work has also been supported by Charles University Research Program No. UNCE/SCI/023.

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Pozza, S., Simoncini, V. Functions of rational Krylov space matrices and their decay properties. Numer. Math. 148, 99–126 (2021). https://doi.org/10.1007/s00211-021-01198-4

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