We introduce an algorithm for estimating the trace of a matrix function $f(\mathbf{A})$ using implicit products with a symmetric matrix $\mathbf{A}$. Existing methods for implicit trace estimation of a matrix function tend to treat matrix-vector products with $f(\mathbf{A})$ as a black-box to be computed by a Krylov subspace method. Like other recent algorithms for implicit trace estimation, our approach is based on a combination of deflation and stochastic trace estimation. However, we take a closer look at how products with $f(\mathbf{A})$ are integrated into these approaches which enables several efficiencies not present in previously studied methods. In particular, we describe a Krylov subspace method for computing a low-rank approximation of a matrix function by a computationally efficient projection onto Krylov subspace.

翻译：我们提出一种算法，用于估计矩阵函数$f(\mathbf{A})$的迹，其中利用对称矩阵$\mathbf{A}$的隐式乘积。现有的矩阵函数隐式迹估计方法通常将$f(\mathbf{A})$的矩阵-向量积视为由Krylov子空间方法计算的黑盒。与其他近期提出的隐式迹估计算法类似，我们的方法基于消元法与随机迹估计的结合。然而，我们深入分析了如何将$f(\mathbf{A})$的乘积整合到这些方法中，从而实现了此前研究中未涉及的多项效率优化。具体而言，我们描述了一种通过计算高效的Krylov子空间投影来构造矩阵函数低秩近似的Krylov子空间方法。