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

  • Ken’ichiro TanakaEmail author
  • Sunao Murashige
Original Paper Area 2


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.


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

Mathematics Subject Classification (2000)

65L15 65L20 65L70 


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© The JJIAM Publishing Committee and Springer Japan 2013

Authors and Affiliations

  1. 1.School of Systems Information ScienceFuture University HakodateHokkaidoJapan

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