Spectrum-Adapted Polynomial Approximation for Matrix Functions
This work addresses the need for efficient matrix function approximations in scientific computing, offering improvements for specific matrix types but with incremental novelty.
The paper proposes two new methods for approximating f(A)b for large sparse Hermitian matrices by first estimating the spectral density and then adapting polynomials to focus on high-density spectral regions. The methods achieve more accurate approximations at lower polynomial orders compared to Lanczos and Chebyshev expansions, particularly for matrices with many distinct interior eigenvalues and small spectral width.
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.