Spectrum-Adapted Polynomial Approximation for Matrix Functions

TL;DR

This paper introduces two novel spectral density-informed polynomial approximation methods for efficiently computing matrix functions times vectors, outperforming existing techniques especially for matrices with complex spectra.

Contribution

The paper presents two new methods that adapt polynomial approximation based on spectral density, improving accuracy and efficiency over traditional approaches.

Findings

More accurate approximations at lower polynomial orders.

Effective for matrices with many interior eigenvalues and narrow spectral width.

Outperforms Lanczos and Chebyshev methods in tested scenarios.

Abstract

We propose and investigate two new methods to approximate $f({\bf A}){\bf b}$ for large, sparse, Hermitian matrices ${\bf A}$. The main idea behind both methods is to first estimate the spectral density of ${\bf A}$, and then find polynomials of a fixed order that better approximate the function $f$ on areas of the spectrum with a higher density of eigenvalues. Compared to state-of-the-art methods such as the Lanczos method and truncated Chebyshev expansion, the proposed methods tend to provide more accurate approximations of $f({\bf A}){\bf b}$ at lower polynomial orders, and for matrices ${\bf A}$ with a large number of distinct interior eigenvalues and a small spectral width.