  The numerical solution of eigenvalue problems is essential in various
application areas of scientific and engineering domains. In many problem
classes, the practical interest is only a small subset of eigenvalues so it is
unnecessary to compute all of the eigenvalues. Notable examples are the
electronic structure problems where the $k$-th smallest eigenvalue is closely
related to the electronic properties of materials. In this paper, we consider
the $k$-th eigenvalue problems of symmetric dense matrices with low-rank
off-diagonal blocks. We present a linear time generalized LDL decomposition of
$\mathcal{H}^2$ matrices and combine it with the bisection eigenvalue algorithm
to compute the $k$-th eigenvalue with controllable accuracy. In addition, if
more than one eigenvalue is required, some of the previous computations can be
reused to compute the other eigenvalues in parallel. Numerical experiments show
that our method is more efficient than the state-of-the-art dense eigenvalue
solver in LAPACK/ScaLAPACK and ELPA. Furthermore, tests on electronic state
calculations of carbon nanomaterials demonstrate that our method outperforms
the existing HSS-based bisection eigenvalue algorithm on 3D problems.
