Solution of the k-th eigenvalue problem in large-scale electronic structure calculations

• Published in: Journal of computational physics Vol. 371; pp. 618 - 632
• Main Authors: Lee, Dongjin, Hoshi, Takeo, Sogabe, Tomohiro, Miyatake, Yuto, Zhang, Shao-Liang
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 15-10-2018; Elsevier Science Ltd
• Subjects: Algorithms; Banach spaces; Computation; Computational physics; Eigenvalues; Eigenvectors; Electronic structure; Electronic structure calculations; Generalized eigenvalue problem; Lanczos method; Material properties; Mathematical analysis; Numerical analysis; Optoelectronic devices; Sparse direct linear solver; Sparse matrices; Sparsity; Spectral bisection; Vector space; Generalized eigenvalue problem; Spectral bisection; Sparse direct linear solver; Lanczos method; Electronic structure calculations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2018.06.002

We consider computing the k-th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of n×n large sparse matrices. In electronic structure calculations, several properties of materials, such as those of optoelectronic device materials, are governed by the eigenpair with a material-specific index k. We present a three-stage algorithm for computing the k-th eigenpair with validation of its index. In the first stage of the algorithm, we propose an efficient way of finding an interval containing the k-th eigenvalue (1≪k≪n) with a non-standard application of the Lanczos method. In the second stage, spectral bisection for large-scale problems is realized using a sparse direct linear solver to narrow down the interval of the k-th eigenvalue. In the third stage, we switch to a modified shift-and-invert Lanczos method to reduce bisection iterations and compute the k-th eigenpair with validation. Numerical results with problem sizes up to 1.5 million are reported, and the results demonstrate the accuracy and efficiency of the three-stage algorithm.