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### Spectral slicing for Hermitian matrices based on Zolotarev’s functions

**PDE and Applied Math Seminar**

Speaker: | Yingzhou Li, Stanford University |

Related Webpage: | http://web.stanford.edu/~ryanlee/ |

Location: | 1147 MSB |

Start time: | Fri, Apr 21 2017, 4:10PM |

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 matricesAandBare of sizeN×NwithO(N)nonzero entries, the computational cost for computingO(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 solvingO(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.