Convergence rates and explicit error bounds of Hill’s method for spectra of self-adjoint differential operators

Original Paper Area 2
  • 77 Downloads

Abstract

We present the convergence rates and the explicit error bounds of Hill’s method, which is a numerical method for computing the spectra of ordinary differential operators with periodic coefficients. This method approximates the operator by a finite dimensional matrix. On the assumption that the operator is self-adjoint, it is shown that, under some conditions, we can obtain the convergence rates of eigenvalues with respect to the dimension and the explicit error bounds. Numerical examples demonstrate that we can verify these conditions using Gershgorin’s theorem for some real problems. Main theorems are proved using the Dunford integrals which project an vector to a specific eigenspace.

Keywords

Hill’s method Convergence rate Error bound Differential operator Eigenvalue problem 

Mathematics Subject Classification (2000)

65L15 65L20 65L70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atkinson K.: The numerical solution of the eigenvalue problem for compact integral operators. Trans. Am. Math. Soc. 129, 458–465 (1967)MATHGoogle Scholar
  2. 2.
    Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 2nd edn. Springer, New York (2005)Google Scholar
  3. 3.
    Curtis C.W., Deconinck B.: On the convergence of Hill’s method. Math. Comput. 79, 169–187 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Dunford, N, Schwartz, J.T.: Linear Operators Part II Spectral Theory, Self Adjoint Operators in Hilbert Space. Wiley, New Jersey (1963)Google Scholar
  5. 5.
    Dunford, N, Schwartz, J.T.: Linear Operators Part III Spectral Operators. Wiley, New Jersey (1971)Google Scholar
  6. 6.
    Deconinck B., Kutz J.N.: Computing spectra of linear operators using Hill’s method. J. Comput. Phys. 219, 296–321 (2006)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hill, G.W.: On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon. Acta Math. 8, 1–36 (1886)Google Scholar
  8. 8.
    Hille, E.: Ordinary Differential Equations in the Complex Domain. Dover, New York (1997)Google Scholar
  9. 9.
    Johnson, M.A., Zumbrun, K. Convergence of Hill’s method for nonselfadjoint operators. SIAM J. Numer. Anal. 50, 64–78 (2012)Google Scholar
  10. 10.
    Kiper A.: Fourier series coefficients for powers of the Jacobian elliptic functions. Math. Comput. 43, 247–259 (1984)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., Rutitskii, Ya.B., Stetsenko, V.Ya.: Approximate Solution of Operator Equations. Wolters-Noordhoff, Groningen (1972)Google Scholar
  12. 12.
    Reed, M., Simon, B.: Methods of modern Mathematical Physics. I. Functional Analysis, revised and enlarged edn. Academic Press, New York (1980)Google Scholar
  13. 13.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)Google Scholar
  14. 14.
    Vainikko, G.M.: A perturbed Galerkin method and the general theory of approximate methods for nonlinear equations. Zh. Vychisl. Mat. Mat. Fiz. 7, 723–751 (1967) (in Russian) (U.S.S.R. Comput. Math. Math. Phys. 7, 18–32 (1967) (in English))Google Scholar
  15. 15.
    Varga, R.S.: Gershgorin and His Circles. Springer, Berlin (2004)Google Scholar
  16. 16.
    Whittaker, E.T., Watson, G.N.: A course of modern analysis, 4th edn. Cambridge University Press, Cambridge (1996) (reprinted)Google Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2013

Authors and Affiliations

  1. 1.School of Systems Information ScienceFuture University HakodateHokkaidoJapan

Personalised recommendations