Advertisement

Inverse Iteration for Nonlinear Eigenvalue Problems in Electromagnetic Scattering

  • A. Spence
  • C. G. Poulton
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 113)

Abstract

We present an extension of the well-known method of “inverse iteration” for the standard eigenvalue problem to the nonlinear problem of finding dispersion relations for electromagnetic waves moving through a doublyperiodic structure. Numerical results are presented to illustrate the performance of the technique. A further improvement is described that allows an e cient “path following” algorithm where a curve of solutions is computed in (ω, Kbloch) space. We present dispersion relations calculated via this new method and compare the efficiency of this algorithm with that of more traditional methods.

Keywords

Eigenvalue Problem Dispersion Curve Search Region Quadratic Convergence Nonlinear Eigenvalue Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    K. M. Leung and Y.F. Liu. Full vector wave calculation of photonic band structures in face-centred cubic dielectric media. Phys. Rev. Lett., 65:2646–2649, 1990.CrossRefGoogle Scholar
  2. [2]
    J. B. Pendry and P. M. Bell. Transfer matrix techniques for electromagnetic waves, volume 315 of NATO ASI Series E: Applied Sciences, page 203. Kluwer, Dordrecht, 1996.Google Scholar
  3. [3]
    R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and Ke-Da Bao. Green’s function, lattice sum and Rayleigh’s identity for a dynamic scattering problem, volume 96 of IMA Volumes in Mathematics and its Applications, pages 155–186. Springer-Verlag, New York, 1997.Google Scholar
  4. [4]
    W. H. Press, S. A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in Fortran 77, section 9.3, CUP, Cambridge MA, 1992.Google Scholar
  5. [5]
    A. Ruhe Algorithms for the Nonlienear Eigenvalue Problem, SIAM J.Numer. Anal. vol 10, pages 674–689. 1973.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    P. M. Anselone and L. B. Rall, The Solution of Characteristic Value-Vector Problems by Newton’s Method, Numerische Mathematik, vol 11, pages 38–45, 1968.CrossRefMathSciNetzbMATHGoogle Scholar
  7. [7]
    P. Lancaster, Lambda-Matrices and Vibrating Systems, Pergamon, Oxford. 1966.zbMATHGoogle Scholar
  8. [8]
    B. N. Parlett, The Symmetric Eigenvalue Problem, 2nd edition, SIAM, Philadelphia, 1998.zbMATHGoogle Scholar
  9. [9]
    P. Lancaster, A Generalised Rayleigh Quotient Iteration for Lambda-Matrices, Arch. Rat. Mech. Anal. vol 8 pages 309–322, 1961.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Neumaier, Residual Inverse Iteration for the Nonlinear Eigenvalue Problem SIAM J.Numer. Anal. vol 20, pages 914–923. 1983.MathSciNetGoogle Scholar
  11. [11]
    J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.zbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • A. Spence
    • 1
  • C. G. Poulton
    • 2
  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.High Frequency and Quantum Electronics LaboratoryUniversity of KarlsruheKarlsruheGermany

Personalised recommendations