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BIT Numerical Mathematics

, Volume 58, Issue 4, pp 907–935 | Cite as

Backward error analysis of polynomial approximations for computing the action of the matrix exponential

  • Marco Caliari
  • Peter Kandolf
  • Franco Zivcovich
Article

Abstract

We describe how to perform the backward error analysis for the approximation of \(\exp (A)v\) by \(p(s^{-1}A)^sv\), for any given polynomial p(x). The result of this analysis is an optimal choice of the scaling parameter s which assures a bound on the backward error, i.e. the equivalence of the approximation with the exponential of a slightly perturbed matrix. Thanks to the SageMath package expbea we have developed, one can optimize the performance of the given polynomial approximation. On the other hand, we employ the package for the analysis of polynomials interpolating the exponential function at so called Leja–Hermite points. The resulting method for the action of the matrix exponential can be considered an extension of both Taylor series approximation and Leja point interpolation. We illustrate the behavior of the new approximation with several numerical examples.

Keywords

Backward error analysis Action of matrix exponential Leja–Hermite interpolation Taylor series 

Mathematics Subject Classification

65D05 65F30 65F60 

References

  1. 1.
    Al-Mohy, A.H., Higham, N.J.: A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. Appl. 31(3), 970–989 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bergamaschi, L., Caliari, M., Vianello, M.: Efficient approximation of the exponential operator for discrete 2D advection–diffusion problems. Numer. Linear Algebra Appl. 10(3), 271–289 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berrut, J.P., Trefethen, N.: Barycentric Lagrange Interpolation. SIAM Rev. 46(3), 501–517 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bos, L.P., Caliari, M.: Application of modified Leja sequences to polynomial interpolation. Dolomites Res. Notes Approx. 8, 66–74 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Caliari, M.: Accurate evaluation of divided differences for polynomial interpolation of exponential propagators. Computing 80(2), 189–201 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Caliari, M., Kandolf, P., Ostermann, A., Rainer, S.: The Leja method revisited: backward error analysis for the matrix exponential. SIAM J. Sci. Comput. 38(3), A1639–A1661 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Caliari, M., Ostermann, A., Rainer, S.: Meshfree exponential integrators. SIAM J. Sci. Comput. 35(1), A431–A452 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Caliari, M., Vianello, M., Bergamaschi, L.: Interpolating discrete advection–diffusion propagators at Leja sequences. J. Comput. Appl. Math. 172(1), 79–99 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Deadman, E.: Estimating the condition number of \(f(A)b\). Numer. Algorithms 70, 287–308 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Druskin, V.L., Knizhnerman, L.A.: Two polynomial methods of calculating functions of symmetric matrices. USSR Comput. Math. Math. Phys. 29(6), 112–121 (1989)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fischer, T.M.: On the algorithm by Al-Mohy and Higham for computing the action of the matrix exponential: a posteriori roundoff error estimation. Linear Algebra Appl. 531, 141–168 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Higham, N.J.: Functions of Matrices. SIAM, Philadelphia (2008)CrossRefGoogle Scholar
  14. 14.
    Higham, N.J., Relton, S.D.: Higher order Fréchet derivatives of matrix functions and the level-2 condition number SIAM. J. Matrix Anal. Appl. 35(3), 1019–1037 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Higham, N.J., Tisseur, F.: A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra. SIAM J. Matrix Anal. Appl. 21(4), 1185–1201 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kalantari, B.: Generalization of Taylor’s theorem and Newton’s method via a new family of determinantal interpolation formulas and its applications. J. Comput. Appl. Math. 126, 287–318 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Luan, V.T., Ostermann, A.: Parallel exponential Rosenbrock methods. Comput. Math. Appl. 71, 1137–1150 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Martínez, A., Bergamaschi, L., Caliari, M., Vianello, M.: A massively parallel exponential integrator for advection–diffusion models. J. Comput. Appl. Math. 231(1), 82–91 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    McCurdy, A.: Accurate computation of divided differences. Technical report, University of California—ERL (1980)Google Scholar
  21. 21.
    McCurdy, A., Ng, K.C., Parlett, B.N.: Accurate computation of divided differences of the exponential function. Math. Comput. 43(168), 501–528 (1984)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Moret, I., Novati, P.: An interpolatory approximation of the matrix exponential based on Faber polynomials. J. Comput. Appl. Math. 131, 361–380 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Novati, P.: A polynomial method based on Fejèr points for the computation of functions of unsymmetric matrices. Appl. Numer. Math. 44, 201–224 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Opitz, V.G.: Steigungsmatrizen. Z. Angew. Math. Mech. 44, T52–T54 (1964)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Reichel, L.: Newton interpolation at Leja points. BIT 30, 332–346 (1990)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schaefer, M.J.: A polynomial based iterative method for linear parabolic equations. J. Comput. Appl. Math. 29, 35–50 (1990)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Stetekluh, J.: http://stetekluh.com/NewtonPoly.html. Accessed 4 Oct 2017
  28. 28.
    Tal-Ezer, H.: High degree polynomial interpolation in Newton form. SIAM J. Sci. Stat. Comput. 12(3), 648–667 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of VeronaVeronaItaly
  2. 2.Department of MathematicsUniversity of InnsbruckInnsbruckAustria
  3. 3.Department of MathematicsUniversity of TrentoTrentoItaly

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