Spectrum-Adapted Polynomial Approximation for Matrix Functions
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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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