Skip to main content

Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices

Abstract

This paper derives a priori residual-type bounds for the Arnoldi approximation of a matrix function together with a strategy for setting the iteration accuracies in the inexact Arnoldi approximation of matrix functions. Such results are based on the decay behavior of the entries of functions of banded matrices. Specifically, a priori decay bounds for the entries of functions of banded non-Hermitian matrices will be exploited, using Faber polynomial approximation. Numerical experiments illustrate the quality of the results.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Afanasjew, M., Eiermann, M., Ernst, O.G., Güttel, S.: Implementation of a restarted Krylov subspace method for the evaluation of matrix functions. Linear Algebra Appl. 429(10), 2293–2314 (2008). https://doi.org/10.1016/j.laa.2008.06.029

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Beckermann, B.: Image numérique, GMRES et polynômes de Faber. C. R. Math. Acad. Sci. Paris 340(11), 855–860 (2005)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Benzi, M., Boito, P.: Decay properties for functions of matrices over \(C^*\)-algebras. Linear Algebra Appl. 456, 174–198 (2014). https://doi.org/10.1016/j.laa.2013.11.027

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Benzi, M., Boito, P., Razouk, N.: Decay properties of spectral projectors with applications to electronic structure. SIAM Rev. 55(1), 3–64 (2013). https://doi.org/10.1137/100814019

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Benzi, M., Golub, G.H.: Bounds for the entries of matrix functions with applications to preconditioning. BIT 39(3), 417–438 (1999). https://doi.org/10.1023/A:1022362401426

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Benzi, M., Razouk, N.: Decay bounds and O(n) algorithms for approximating functions of sparse matrices. Electron. Trans. Numer. Anal. 28, 16–39 (2007)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Benzi, M., Simoncini, V.: Decay bounds for functions of Hermitian matrices with banded or Kronecker structure. SIAM J. Matrix Anal. Appl. 36(3), 1263–1282 (2015). https://doi.org/10.1137/151006159

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Botchev, M.A., Grimm, V., Hochbruck, M.: Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35(3), A1376–A1397 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Canuto, C., Simoncini, V., Verani, M.: On the decay of the inverse of matrices that are sum of Kronecker products. Linear Algebra Appl. 452, 21–39 (2014). https://doi.org/10.1016/j.laa.2014.03.029

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Cowen, C.C., Harel, E.: An Effective Algorithm for Computing the Numerical Range (1995). https://www.math.iupui.edu/ccowen/Downloads/33NumRange.html

  11. 11.

    Del Buono, N., Lopez, L., Peluso, R.: Computation of the exponential of large sparse skew-symmetric matrices. SIAM J. Sci. Comput. 27(1), 278–293 (2005). https://doi.org/10.1137/030600758

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Demko, S.G.: Inverses of band matrices and local convergence of spline projections. SIAM J. Numer. Anal. 14(4), 616–619 (1977). https://doi.org/10.1137/0714041

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Demko, S.G., Moss, W.F., Smith, P.W.: Decay rates for inverses of band matrices. Math. Comp. 43(168), 491–499 (1984). https://doi.org/10.2307/2008290

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Dinh, K.N., Sidje, R.B.: Analysis of inexact Krylov subspace methods for approximating the matrix exponential. Math. Comput. Simul. 138, 1–13 (2017). https://doi.org/10.1016/j.matcom.2017.01.002. http://www.sciencedirect.com/science/article/pii/S0378475417300034

    MathSciNet  Article  Google Scholar 

  15. 15.

    Druskin, V., Knizhnerman, L.: Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic. Numer. Linear Algebra Appl. 2(3), 205–217 (1995). https://doi.org/10.1002/nla.1680020303

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Eiermann, M.: Fields of values and iterative methods. Linear Algebra Appl. 180, 167–197 (1993). https://doi.org/10.1016/0024-3795(93)90530-2

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Eiermann, M., Ernst, O.G., Güttel, S.: Deflated restarting for matrix functions. SIAM J. Matrix Anal. Appl. 32(2), 621–641 (2011)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Eijkhout, V., Polman, B.: Decay rates of inverses of banded M-matrices that are near to Toeplitz matrices. Linear Algebra Appl. 109, 247–277 (1988). https://doi.org/10.1016/0024-3795(88)90211-X. http://www.sciencedirect.com/science/article/pii/002437958890211X

    MathSciNet  Article  Google Scholar 

  19. 19.

    Ellacott, S.W.: Computation of Faber series with application to numerical polynomial approximation in the complex plane. Math. Comp. 40(162), 575–587 (1983)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Freund, R.: On polynomial approximations to \(f_a(z)=(z-a)^{-1}\) with complex \(a\) and some applications to certain non-hermitian matrices. Approx. Theory Appl. 5, 15–31 (1989)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Frommer, A., Güttel, S., Schweitzer, M.: Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices. SIAM J. Matrix Anal. Appl. 35(4), 1602–1624 (2014)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Frommer, A., Güttel, S., Schweitzer, M.: Efficient and stable Arnoldi restarts for matrix functions based on quadrature. SIAM J. Matrix Anal. Appl. 35(2), 661–683 (2014). https://doi.org/10.1137/13093491X

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Frommer, A., Schimmel, C., Schweitzer, M.: Bounds for the decay of the entries in inverses and Cauchy-Stieltjes functions of certain sparse, normal matrices. Numer. Linear Algebra Appl. 25(4), e2131 (2018). https://doi.org/10.1002/nla.2131

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Giraud, L., Langou, J., Rozložník, M., van den Eshof, J.: Rounding error analysis of the classical Gram-Schmidt orthogonalization process. Numer. Math. 101(1), 87–100 (2005)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Güttel, S.: Rational Krylov approximation of matrix functions: numerical methods and optimal pole selection. GAMM-Mitt. 36(1), 8–31 (2013). https://doi.org/10.1002/gamm.201310002

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Güttel, S., Knizhnerman, L.: A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions. BIT 53(3), 595–616 (2013). https://doi.org/10.1007/s10543-013-0420-x

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia (2008)

    Book  Google Scholar 

  28. 28.

    Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Iserles, A.: How large is the exponential of a banded matrix? New Zealand J. Math. 29, 177–192 (2000)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Knizhnerman, L., Simoncini, V.: A new investigation of the extended Krylov subspace method for matrix function evaluations. Numer. Linear Algebra Appl. 17(4), 615–638 (2010). https://doi.org/10.1002/nla.652

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Kürschner, P., Freitag, M.A.: Inexact methods for the low rank solution to large scale Lyapunov equations. arXiv preprint arXiv:1809.06903 (2018)

  32. 32.

    Matrix Market: A Visual Repository of Test Data for Use in Comparative Studies of Algorithms for Numerical Linear Algebra. Mathematical and Computational Sciences Division, National Institute of Standards and Technology; available online at http://math.nist.gov/MatrixMarket

  33. 33.

    Mastronardi, N., Ng, M., Tyrtyshnikov, E.E.: Decay in functions of multiband matrices. SIAM J. Matrix Anal. Appl. 31(5), 2721–2737 (2010). https://doi.org/10.1137/090758374

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    The MathWorks, Inc.: MATLAB 7, r2013b edn. (2013)

  35. 35.

    Meurant, G.: A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM J. Matrix Anal. Appl. 13(3), 707–728 (1992). https://doi.org/10.1137/0613045

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Pozza, S., Tudisco, F.: On the stability of network indices defined by means of matrix functions. SIAM J. Matrix Anal. Appl. 39(4), 1521–1546 (2018)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Rinehart, R.F.: The equivalence of definitions of a matric function. Amer. Math. Monthly 62(6), 395–414 (1955)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Saad, Y.: Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29(1), 209–228 (1992). https://doi.org/10.1137/0729014

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Simoncini, V.: Variable accuracy of matrix-vector products in projection methods for eigencomputation. SIAM J. Numer. Anal. 43(3), 1155–1174 (2005)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Simoncini, V., Szyld, D.B.: Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM J. Sci. Comput. 25(2), 454–477 (2003)

    MathSciNet  Article  Google Scholar 

  41. 41.

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

  42. 42.

    Wang, H.: The Krylov Subspace Methods for the Computation of Matrix Exponentials. Ph.D. thesis, Department of Mathematics, University of Kentucky (2015)

  43. 43.

    Wang, H., Ye, Q.: Error bounds for the Krylov subspace methods for computations of matrix exponentials. SIAM J. Matrix Anal. Appl. 38(1), 155–187 (2017). https://doi.org/10.1137/16M1063733

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Ye, Q.: Error bounds for the Lanczos methods for approximating matrix exponentials. SIAM J. Numer. Anal. 51(1), 68–87 (2013). https://doi.org/10.1137/11085935x

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

We are indebted with Leonid Knizhnerman for a careful reading of an earlier version of this manuscript, and for his many insightful remarks which led to great improvements of our results. We also thank Michele Benzi for several suggestions and the two referees whose remarks helped us improve the presentation.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Stefano Pozza.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been supported by the FARB12SIMO grant, Università di Bologna, by INdAM-GNCS under the 2016 Project Equazioni e funzioni di matrici con struttura: analisi e algoritmi, by the INdAM-GNCS “Giovani ricercatori 2016” grant, and by Charles University Research program No. UNCE/SCI/023.

Communicated by Daniel Kressner.

Appendix. Technical proofs

Appendix. Technical proofs

Proof of corollary 2.4

Let \(\rho = \sqrt{a^2 - b^2}\) be the distance between the foci and the center of the ellipse (i.e., the boundary of E), and let \(R = (a + b)/\rho \). Then a conformal map for E is

$$\begin{aligned} \phi (w) = \frac{w - c - \sqrt{(w-c)^2 - \rho ^2 }}{\rho R}, \end{aligned}$$
(A.1)

and its inverse is

$$\begin{aligned} \psi (z) = \frac{\rho }{2}\left( Rz + \frac{1}{Rz} \right) + c \,; \end{aligned}$$
(A.2)

see, e.g., [41, chapter II, Example 3]. Notice that

$$\begin{aligned} \max _{|z| = \tau } |e^{\psi (z)}| = \max _{|z| = \tau } e^{\mathfrak {R}(\psi (z))} = e^{\frac{\rho }{2}\left( R\tau + \frac{1}{R\tau } \right) + c_1}. \end{aligned}$$

Hence by Theorem 2.3 we get

$$\begin{aligned} \left| \left( e^{A} \right) _{k,\ell } \right| \le 2 \frac{\tau }{\tau - 1} e^{c_1} e^{\frac{\rho }{2}\left( R\tau + \frac{1}{R\tau } \right) } \left( \frac{1}{\tau } \right) ^\xi . \end{aligned}$$

The optimal value of \(\tau > 1\) that minimizes \( e^{\frac{\rho }{2}\left( R\tau + \frac{1}{R\tau } \right) } \left( \frac{1}{\tau } \right) ^\xi \) is

$$\begin{aligned} \tau = \frac{\xi + \sqrt{\xi ^2 + \rho ^2 }}{\rho R}. \end{aligned}$$

Moreover, the condition \(\tau >1\) is satisfied if and only if \(\xi > \frac{\rho }{2}\left( R - \frac{1}{R} \right) = b\). Finally, noticing that

$$\begin{aligned} \psi \left( \frac{\xi + \sqrt{\xi ^2 + \rho ^2 }}{\rho R}\right) - c_1 = \frac{1}{2}\left( \xi + \sqrt{\xi ^2 + \rho ^2 } + \frac{\rho ^2}{\xi + \sqrt{\xi ^2 + \rho ^2 }} \right) = \xi q(\xi ), \end{aligned}$$

and collecting \(\xi \) the proof is completed. \(\square \)

Proof of Corollary 2.5

The function \(f(z) = \exp (- \sqrt{z})\) is analytic on \(\mathbb {C} \setminus (-\infty ,0)\). Since we consider the principal square root, then \(\mathfrak {R}(\sqrt{z})\ge 0\), and

$$\begin{aligned} |\exp (- \sqrt{z})| = \exp (-\mathfrak {R}(\sqrt{z})) \le 1. \end{aligned}$$

Hence, by Theorem 2.3 we can determine \(\tau \) for which

$$\begin{aligned} \left| \left( e^{-\sqrt{A}} \right) _{k,\ell } \right| \le 2 \frac{\tau }{\tau - 1} \left( \frac{1}{\tau } \right) ^\xi . \end{aligned}$$

For every \(\varepsilon > 0\) close enough to zero, we set the parameter

$$\begin{aligned} \tau _\varepsilon = |\phi (\varepsilon )| = \left| \frac{c - \varepsilon + \sqrt{(c-\varepsilon )^2 -\rho ^2}}{\rho R} \right| , \end{aligned}$$

with \(\phi (w)\) as in (A.1) and \(\psi (z)\) its inverse (A.2). Then the ellipse \(\{ \psi (z), \, |z| = \tau _\varepsilon \}\) is contained in \(\mathbb {C}{\setminus }(-\infty ,0]\). Letting \(\varepsilon \rightarrow 0\) concludes the proof. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pozza, S., Simoncini, V. Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices. Bit Numer Math 59, 969–986 (2019). https://doi.org/10.1007/s10543-019-00763-6

Download citation

Keywords

  • Arnoldi algorithm
  • Inexact Arnoldi algorithm
  • Matrix functions
  • Faber polynomials
  • Decay bounds
  • Banded matrices

Mathematics Subject Classification

  • 65F60
  • 65F10