Eigenvalue solvers for three dimensional photonic crystals with face-centered cubic lattice

Tsung Ming Huang, Han En Hsieh, Wen Wei Lin, Weichung Wang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


To numerically determine the band structure of three-dimensional photonic crystals with face-centered cubic lattices, we study how the associated large-scale generalized eigenvalue problem (GEP) can be solved efficiently. The main computational challenge is due to the complexity of the coefficient matrix and the fact that the desired eigenvalues are interior. For solving the GEP by the shift-and-invert Lanczos method, we propose a preconditioning for the associated linear system therein. Recently, a way to reformat the GEP to the null space free eigenvalue problem (NFEP) is proposed. For solving the NFEP, we analyze potential advantages and disadvantages of the null space free inverse Lanczos method, the shift-invert residual Arnoldi method, and the Jacobi-Davidson method from theoretical viewpoints. These four approaches are compared numerically to find out their properties. The numerical results suggest that the shift-invert residual Arnoldi method with an initialization scheme is the fastest and the most robust eigensolver for the target eigenvalue problems. Our findings promise to play an essential role in simulating photonic crystals.

Original languageEnglish
Pages (from-to)350-361
Number of pages12
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - 2014 Dec 15


  • Face-centered cubic lattice
  • Fast Fourier transform matrix-vector multiplications
  • Maxwell's equations
  • Null space free eigenvalue problem
  • Shift-invert residual Arnoldi method
  • Three-dimensional photonic crystals

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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