Advertisement

On a Generalization of Neumann Series of Bessel Functions Using Hessenberg Matrices and Matrix Exponentials

  • A. Koskela
  • E. JarlebringEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

The Neumann expansion of Bessel functions (of integer order) of a function \(g:\mathbb {C}\rightarrow \mathbb {C}\) corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., \(g(s)=\sum _{\ell = 0}^\infty w_\ell \varphi _\ell (s)\), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

References

  1. 1.
    M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Applied Mathematics Series, vol. 55 (National Bureau of Standards, Washington, 1964)Google Scholar
  2. 2.
    M. Benzi, N. Razouk, Decay bounds and O(n) algorithms for approximating functions of sparse matrices. Electron. Trans. Numer. Anal. 28, 16–39 (2007)Google Scholar
  3. 3.
    A. Iserles, How large is the exponential of a banded matrix? N. Z. J. Math. 29(2), 177192 (2000)Google Scholar
  4. 4.
    F.H. Jackson, A generalization of Neumann’s expansion of an arbitrary function in a series of Bessel’s functions. Proc. Lond. Math. Soc. s2-1(1), 361–366 (1904)Google Scholar
  5. 5.
    D. Jankov, T.K. Pogány, E. Süli, On the coefficients of Neumann series of Bessel functions. J. Math. Anal. Appl. 380(2), 628–631 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    E. Jarlebring, K. Meerbergen, W. Michiels, A Krylov method for the delay eigenvalue problem. SIAM J. Sci. Comput. 32(6), 3278–3300 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Koskela, E. Jarlebring, The infinite Arnoldi exponential integrator for linear inhomogeneous ODEs. Technical Report. KTH Royal Institute of Technology (2015). http://arxiv.org/abs/1502.01613
  8. 8.
    A. Koskela, E. Jarlebring, M.E. Hochstenbach, Krylov approximation of linear ODEs with polynomial parameterization.. SIAM J. Matrix Anal. Appl. 37(2), 519–538 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    C. Moler, C.V. Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge, 1995)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.KTH Royal Institute of TechnologyStockholmSweden

Personalised recommendations