Singular value decompositions for single-curl operators in three-dimensional Maxwell's equations for complex media

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

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


This article focuses on solving the generalized eigenvalue problems (GEP) arising in the source-free Maxwell equation with magnetoelectric coupling effects that models three-dimensional complex media. The goal is to compute the smallest positive eigenvalues, and the main challenge is that the coefficient matrix in the discrete Maxwell equation is indefinite and degenerate. To overcome this difficulty, we derive a singular value decomposition (SVD) of the discrete single-curl operator and then explicitly express the basis of the invariant subspace corresponding to the nonzero eigenvalues of the GEP. Consequently, we reduce the GEP to a null space free standard eigenvalue problem (NFSEP) that contains only the nonzero (complex) eigenvalues of the GEP and can be solved by the shift-and-invert Arnoldi method without being disturbed by the null space. Furthermore, the basis of the eigendecomposition is chosen carefully so that we can apply fast Fourier transformation (FFT-) based matrix vector multiplication to solve the embedded linear systems efficiently by an iterative method. For chiral and pseudochiral complex media, which are of great interest in magnetoelectric applications, the NFSEP can be further transformed to a null space free GEP whose coefficient matrices are Hermitian and Hermitian positive definite (HHPD-NFGEP). This HHPD-NFGEP can be solved by using the invert Lanczos method without shifting. Furthermore, the embedded linear system can be solved efficiently by using the conjugate gradient method without preconditioning and the FFT-based matrix vector multiplications. Numerical results are presented to demonstrate the efficiency of the proposed methods.

Original languageEnglish
Pages (from-to)203-224
Number of pages22
JournalSIAM Journal on Matrix Analysis and Applications
Issue number1
Publication statusPublished - 2015


  • Chiral medium
  • Discrete single-curl operator
  • Null space free method
  • Pseudochiral medium
  • Singular value decomposition
  • The Maxwell equations

ASJC Scopus subject areas

  • Analysis


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