Efficient estimation of eigenvalue counts in an interval
For applications requiring eigenvalue counts in large sparse matrices, this work offers improved estimation methods that balance accuracy and computational cost.
The paper reviews and proposes new techniques for estimating eigenvalue counts in intervals of large sparse Hermitian matrices, combining polynomial and rational approximation filtering with stochastic methods to provide rough approximations efficiently.
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.