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This work 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 × 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 − σB)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.

This is joint work with Haizhao Yang.