Spectrum-adapted Polynomial Approximation for Matrix Functions

Abstract

We propose and investigate two new methods to approximate f (A)b for large, sparse, Hermitian matrices A. The main idea behind both methods is to first estimate the spectral density of 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 (A)b at lower polynomial orders, and for matrices A with a large number of distinct interior eigenvalues and a small spectral width.

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“…second option for mitigating the approximation error is to choose polynomials that control the error in specific parts of the spectrum, such as transformed linear phase multirate filters [23], which reduce the error near the eigenvalue 0 (no DC leakage) or spectrum-adapted polynomial approximation [69], which can reduce the error in high density areas of the spectrum. A third option is to directly choose the initial set of filters to be polynomials, or at least choose them to be smoother functions that are more accurately approximated by polynomials (e.g., [20]).…”