Title: Interior Eigensolver for Sparse Hermitian Definite Matrices Based on Zolotarev's Functions,
Abstract: This paper proposes an efficient method for computing selected generalized
eigenpairs of a sparse Hermitian definite matrix pencil (A, B). Based on
Zolotarev's best rational function approximations of the signum function and
conformal mapping techniques, we construct the best rational function
approximation of a rectangular function supported on an arbitrary interval.
This new best rational function approximation is applied to construct spectrum
filters of (A, B). Combining fast direct solvers and the shift-invariant GMRES,
a hybrid fast algorithm is proposed to apply spectral filters efficiently.
Compared to the state-of-the-art algorithm FEAST, the proposed rational
function approximation is proved to be optimal among a larger function class,
and the numerical implementation of the proposed method is also faster. The
efficiency and stability of the proposed method are demonstrated by numerical
examples from computational chemistry.