Filter analysis for the stochastic estimation of eigenvalue counts
Yasuyuki Maeda, Yasunori Futamura, Akira Imakura, Tetsuya Sakurai · JSIAM Letters · 2015
To estimate the number of eigenvalues of a Hermitian matrix that are located in a given interval, existing methods include polynomial filtering and rational filtering. Both filtering approaches are based on stochastic approximations for matrix trace. In this paper, we analyze a rational filtering method that is based on polynomial filtering in which the solutions to the linear systems are approximated by a Krylov subspace method. Our analysis and numerical experiments indicate that the rational filtering method is effective when the eigenvalues of a given matrix are sparsely distributed in the target interval.