Efficient estimation of eigenvalue counts in an interval

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-andconquer 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. We also discuss how the latter method is particularly wellsuited for the FEAST eigensolver.ESTIMATING EIGENVALUE COUNTS 675 count OEa; b in OEa; b. While this method yields an exact count, it requires two complete LDL T factorizations, and this can be quite expensive for realistic eigenproblems.This paper discusses two alternative methods that provide only an estimate for OEa; b , but which are relatively inexpensive. Both methods work by estimating the trace of the spectral projector P associated with the eigenvalues inside the interval OEa; b. This spectral projector is expanded in two different ways, and its trace is computed by resorting to stochastic trace estimators, for example, [10,11]. The first method utilizes filtering techniques based on Chebyshev polynomials. The resulting projector is expanded as a polynomial function of A. In the second method, the projector is constructed by integrating the resolvent of the eigenproblem along a contour in the complex plane enclosing the interval OEa; b. In this case, the projector is approximated by a rational function of A.For each of the aforementioned methods, we present various implementations depending on the nature of the eigenproblem (generalized versus standard) and cost considerations. Thus, in the polynomial expansion case, we propose a barrier-type filter when dealing with a standard eigenproblem and two high/low pass filters in the case of generalized eigenproblems. In the rational expansion case, we have the choice of using an LU factorization or a Krylov subspace method to solve linear systems. The optimal implementation of each method used for the eigenvalue count depends on the situation at hand and involves compromises between cost and accuracy. While it is not the aim of this paper to explore detailed analysis of these techniques, we will discuss various possibilities and provide illustrative examples.The polynomial and rational expansion methods are motivated by two distinct approaches recently suggested in the context of electronic structure calculations: (i) spectrum slicing and (ii) Cauchy integral eigen-projection. In the spectrum slicing techniques [1], the eigenpairs are computed by dividing the spectrum in many small subintervals, called 'slices' or 'windows'. For each window, a barrier function is approximated by Chebyshev-Jackson polynomials in order to select only the portion of the spectrum in the slice. In this method, it is important to determine an approximate c...

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