We find formulas of the Chebyshev-polynomial expansion of the energy dependence of the inverse localization length of Hermitian and non-Hermitian random chains. For Hermitian models, the expansion numerically produces the energy dependence as a function in one run of the algorithm. This is in strong contrast to the standard transfer-matrix method, which produces the inverse localization length for a fixed energy in each run. For non-Hermitian models, our algorithm also produces the inverse localization length for a fixed (complex) energy. We also find a formula of the Chebyshev-polynomial expansion of the density of states of non-Hermitian models. In all cases, we do not need to store the entire matrix in computer memory because the algorithm is based on the multiplication of the Hamiltonian matrix. Our algorithm for non-Hermitian models may be the only available one that enables us to find the density of states of models with interactions.

Naomichi Hatano got his Ph.D. in 1993 from University of Tokyo under supervision of Prof. Masuo Suzuki. He has then worked in Harvard University, Los Alamos National Laboratory, Aoyama Gakuin University, and now at Institute of Industrial Science, University of Tokyo. He has worked in various topics of statistical and condensed-matter physics, including non-Hermitian quantum systems, open quantum systems, spin-orbit interaction, quantum Monte Carlo methods and complex networks. He loves collaborating on physics with friends world-wide.