To solve the nonlinear eigenvalue problem, we employ a Newton-type iterative method. To compute successive eigenvalue–eigenvector pairs, we introduce a novel nonequivalence deflation technique that shifts converged eigenvalues to infinity while leaving the remaining spectrum unaffected. The deflated problem is then solved using the same Newton-type method, which serves as a hybrid approach—integrating features of the Jacobi–Davidson and nonlinear Arnoldi methods—to compute clustered eigenvalues effectively.
Example: The numerical simulation of band structures in three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices results in large-scale nonlinear eigenvalue problems. These problems are particularly challenging due to the presence of a high-dimensional eigenspace corresponding to the zero eigenvalue, and the fact that the target eigenvalues—those with the smallest real parts—are tightly clustered and located near the zero eigenvalues.
MATLAB code for the example: main_tmp.m
Download (NLEP_metallic_2014_0320.zip)
License: Copyright (c) 2025 by Tsung-Ming Huang, Wen-Wei Lin and Volker Mehrmann
Reference:
Tsung-Ming Huang, Wen-Wei Lin, and Volker Mehrmann, A Newton-type method with nonequivalence deflation for nonlinear eigenvalue problems arising in photonic crystal modeling, SIAM J. Sci. Comput. Vol. 38, No. 2, B191-B218, 2016.