主講人：Professor Chun-Hua Guo (Department of Mathematics and Statistics, University of Regina, Canada)

主題：Detecting and solving overdamped quadratic eigenvalue problems

Abstract：
Overdamped quadratic matrix polynomials Q(_) = _2A+_B+C are an im- portant class of Hermitian matrix polynomials with nonpositive real eigenvalues. Neither the de_nition of overdamped nor any of the standard characterizations provides an e_cient way to test if a given Q has this property. We show that a matrix iteration based on cyclic reduction provides necessary and su_cient con- ditions for Q to be overdamped. The convergence of this iteration is quadratic if Q is indeed overdamped. For weakly overdamped Q the iteration is shown to be generically linearly convergent with rate at worst 1/2. We show that the matrix iteration can be implemented in such a way that when overdamping is detected a scalar _ < 0 is provided that lies in the gap between the n largest and n small- est eigenvalues of the n _ n quadratic eigenvalue problem (QEP) Q(_)x = 0. Once such a _ is known, the QEP can be solved by linearizing to a de_nite pencil that can be reduced using already available Cholesky factorizations to a standard Hermitian eigenproblem. We obtain an e_cient algorithm that iden- ti_es and solves an overdamped QEP maintaining symmetry throughout and guaranteeing real computed eigenvalues.

時間：2009 年 4 月 27 日13:20~14:10

地點：新生大樓202 室