Spectrum Slicing 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 maps, 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. Assuming that the sparse Hermitian matrices $A$ and $B$ are of size $N\times N$ with $O(N)$ nonzero entries, the computational cost for computing $O(1)$ interior eigenpairs is bounded by that of solving a shifted linear system $(A-\sigma B)x=b$. Utilizing the spectrum slicing idea, the proposed method computes the full eigenvalue decomposition of a sparse Hermitian definite matrix pencil via solving $O(N)$ linear systems. The efficiency and stability of the proposed method are demonstrated by numerical examples of a wide range of sparse matrices. Compared with existing spectrum slicing algorithms based on contour integrals, the proposed method is faster and more reliable.