Interior eigensolver 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 the signum and conformal mapping techniques, we construct approximation rectangular supported arbitrary interval via compositions with partial fraction representations. new can be applied to spectrum filters $(A,B)$ smaller number poles than direct construction without compositions. Combining fast solvers shift-invariant minimal residual method, hybrid algorithm is proposed apply spectral efficiently. Compared state-of-the-art FEAST, more when factorizations are required solve multi-shift linear systems in eigensolver, since needed our method. The efficiency stability demonstrated by numerical examples from computational chemistry.