We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix A, defined as \({{\,\textrm{tr}\,}}(f(A))\) 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.

我们考虑逼近大型、稀疏、对称正半定矩阵A的冯诺依曼熵的问题，定义为\({{\,\textrm{tr}\,}}(f(A))\)其中\( f(x)=-x\log x\)。在建立了该矩阵函数的一些有用属性之后，我们考虑在两种类型的近似方法中使用多项式和有理 Krylov 子空间算法，即随机迹估计器和基于图着色的探测技术。我们开发了用于算法实现的错误界限和启发式方法。对不同类型网络的密度矩阵的数值实验说明了该方法的性能。