Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solution of the $k$-th eigenvalue problem in large-scale electronic
  structure calculations   We consider computing the $k$-th eigenvalue and its corresponding eigenvector
of a generalized Hermitian eigenvalue problem of $n\times n$ 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 $(1 \ll k \ll n)$ 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.

13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph