Spectrum Slicing for Sparse Hermitian Definite Matrices Based on Zolotarev's Functions

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.