Approximating Spectral Densities of Large Matrices

8 years 29 days ago

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In physics, it is sometimes desirable to compute the so-called Density Of States (DOS), also known as the spectral density, of a Hermitian (or symmetric) matrix A. The spectral density can be viewed as a probability density distribution that measures the likelihood of ﬁnding eigenvalues near some point on the real line. The most straightforward way to obtain this density is to compute all eigenvalues of A. But this approach is generally costly and wasteful, especially for matrices of large dimension. There exist alternative methods that allow us to estimate the spectral density function at much lower cost. The major computational cost of these methods is in multiplying A with a number of vectors, which makes them appealing for large-scale problems where products of the matrix A with arbitrary vectors are inexpensive. This paper deﬁnes the problem of estimating the spectral density carefully. It then surveys a few known methods for estimating the spectral density, and proposes some ...

Lin Lin, Yousef Saad, Chao Yang

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