Efficient estimation of eigenvalue counts in an interval

TL;DR

This paper reviews and explores new stochastic and approximation filtering techniques for efficiently estimating the number of eigenvalues within a specific interval of large sparse Hermitian matrices, crucial for divide-and-conquer eigensolvers.

Contribution

It introduces novel polynomial and rational approximation filtering methods combined with stochastic procedures for eigenvalue count estimation.

Findings

New filtering techniques improve estimation efficiency

Stochastic methods provide rough eigenvalue counts quickly

Review of standard approaches contextualizes new methods

Abstract

Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications and it is a prerequisite of eigensolvers based on a divide-and-conquer paradigm. Often an exact count is not necessary and methods based on stochastic estimates can be utilized to yield rough approximations. This paper examines a number of techniques tailored to this specific task. It reviews standard approaches and explores new ones based on polynomial and rational approximation filtering combined with a stochastic procedure.