  We consider the problem of approximating the von Neumann entropy of a large,
sparse, symmetric positive semidefinite matrix $A$, defined as
$\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful
properties of this matrix function, we consider the use of both polynomial and
rational Krylov subspace algorithms within two types of approximations methods,
namely, randomized trace estimators and probing techniques based on graph
colorings. We develop error bounds and heuristics which are employed in the
implementation of the algorithms. Numerical experiments on density matrices of
different types of networks illustrate the performance of the methods.
