Papers › Solution of the k-th eigenvalue problem in large-scale electronic structure calculations
Solution of the k-th eigenvalue problem in large-scale electronic structure calculations
Dongjin Lee, Takeo Hoshi, Tomohiro Sogabe, Yuto Miyatake, Shao-Liang Zhang
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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.
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