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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 usefulproperties of this matrix function, we consider the use of both polynomial andrational Krylov subspace algorithms within two types of approximations methods,namely, randomized trace estimators and probing techniques based on graphcolorings. We develop error bounds and heuristics which are employed in theimplementation of the algorithms. Numerical experiments on density matrices ofdifferent types of networks illustrate the performance of the methods.

Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods