Communications in Mathematical Sciences

Volume 19 (2021)

Number 4

Interior eigensolver for sparse Hermitian definite matrices based on Zolotarev’s functions

Pages: 1113 – 1135



Yingzhou Li (School of Mathematical Sciences, Fudan University, Shanghai, China)

Haizhao Yang (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


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 via function compositions with partial fraction representations. This new best rational function approximation can be applied to construct spectrum filters of $(A,B)$ with a smaller number of poles than a direct construction without function compositions. Combining fast direct solvers and the shift-invariant generalized minimal residual method, 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 more efficient when sparse matrix factorizations are required to solve multi-shift linear systems in the eigensolver, since a smaller number of matrix factorizations is needed in our method. The efficiency and stability of the proposed method are demonstrated by numerical examples from computational chemistry.


generalized eigenvalue problem, spectrum slicing, rational function approximation, sparse Hermitian matrix, Zolotarev’s function, shift-invariant GMRES

2010 Mathematics Subject Classification

44A55, 65R10, 65T50

Received 28 February 2019

Accepted 19 December 2020

Published 18 June 2021