  In physics, it is sometimes desirable to compute the so-called \emph{Density
Of States} (DOS), also known as the \emph{spectral density}, of a real
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 exists
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 relatively inexpensive. This paper defines the problem of estimating the
spectral density carefully, and discusses how to measure the accuracy of an
approximate spectral density. It then surveys a few known methods for
estimating the spectral density, and proposes some new variations of existing
methods. All methods are discussed from a numerical linear algebra point of
view.
