Fast computation of spectral densities for generalized eigenvalue problems

TL;DR

This paper introduces two efficient methods based on polynomial approximation and Lanczos quadrature to estimate spectral densities of large Hermitian matrix pencils, enabling spectrum analysis without explicit eigenvalue computation.

Contribution

It presents novel algorithms using Chebyshev polynomial techniques for spectral density estimation of Hermitian matrix pencils, with convergence analysis and practical spectrum slicing applications.

Findings

Lanczos method converges twice as fast as KPM under certain conditions.

Lanczos provides more accurate spectral densities for non-uniform eigenvalue distributions.

The methods enable spectrum partitioning for large matrices without explicit eigenvalue calculation.

Abstract

The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics, or a social network in behavioral sciences. However, computing all the eigenvalues explicitly is prohibitively expensive for real-world applications. This paper presents two types of methods to efficiently estimate the spectral density of a matrix pencil $(A, B)$ when both $A$ and $B$ are Hermitian and, in addition, $B$ is positive definite. The first one is based on the Kernel Polynomial Method (KPM) and the second on Gaussian quadrature by the Lanczos procedure. By employing Chebyshev polynomial approximation techniques, we can avoid direct factorizations in both methods, making the resulting algorithms suitable for large matrices. Under some assumptions, we prove bounds that suggest that the Lanczos method converges twice as fast as the KPM method. Numerical examples further indicate that the Lanczos method can provide more accurate spectral densities when the eigenvalue distribution is highly non-uniform. As an application, we show how to use the computed spectral density to partition the spectrum into intervals that contain roughly the same number of eigenvalues. This procedure, which makes it possible to compute the spectrum by parts, is a key ingredient in the new breed of eigensolvers that exploit "spectrum slicing".