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.
翻译:我们考虑的是,对于一个庞大的、稀疏的、对称正正半确定基质的半确定基质(A)美元,被定义为$\Operatorname{tr}(f(A))美元,即$f(x)=-x\logx$(F(A))美元,而$f(x)=-x\logx$(美元)美元则定义为$f(x)=-xxx$。在确定该基质函数的一些有用属性之后,我们考虑在两种近似方法中使用多元和理性的Krylov亚空间算法,即随机的微量测算器和基于图表颜色的探测技术。我们开发了在实施算法时使用的误差界限和超常度。关于不同类型网络密度矩阵的数值实验说明了方法的性能。