A Contour-Integral Based Method for Counting the Eigenvalues Inside a Region

Abstract

In many applications, the information about the number of eigenvalues inside a given region is required. In this paper, we propose a contour-integral based method for this purpose. The new method is motivated by two findings. There exist methods for estimating the number of eigenvalues inside a region in the complex plane. But our method is able to compute the number of eigenvalues inside the given region exactly. An appealing feature of our method is that it can integrate with the recently developed contour-integral based eigensolvers to help them detect whether all desired eigenvalues are found. Numerical experiments are reported to show the viability of our new method.

A spectral projection method for transmission eigenvalues

“…Randomly chosen functions are projected to the generalized eigenspace corresponding to the eigenvalues inside a closed contour, which leads to a relative small finite dimension eigenvalue problem. For recently developments along this line, we refer the readers to [22,33,32,31] For most existing integral based methods, estimation on the locations, number of eigenvalues and dimensions of eigenspace are critical for their successes. The proposed method is related to the methods developed in [21] and [14].…”

An algorithm for identifying eigenvectors exhibiting strong spatial localization

“…One such variant is to first estimate (as efficiently as possible) how many eigenvalues of L are "near" µ 1 (cf. [14,50]), as this has a direct affect on the convergence rate of inverse iteration. Recall that we are guaranteed that there is at least one eigenvalue of L that is within sδ(φ, R)τ (φ, R) of µ 1 , so we might consider a slightly larger interval around µ 1 for our eigenvalue count estimate.…”