In this paper, we explore quadratures for the evaluation of $B^T \phi(A) B$ where $A$ is a symmetric nonnegative-definite matrix in $\mathbb{R}^{n \times n}$, $B$ is a tall matrix in $\mathbb{R}^{n \times p}$, and $\phi(\cdot)$ represents a matrix function that is regular enough in the neighborhood of $A$'s spectrum, e.g., a Stieltjes or exponential function. These formulations, for example, commonly arise in the computation of multiple-input multiple-output (MIMO) transfer functions for diffusion PDEs. We propose an approximation scheme for $B^T \phi(A) B$ leveraging the block Lanczos algorithm and its equivalent representation through Stieltjes matrix continued fractions. We extend the notion of Gauss-Radau quadrature to the block case, facilitating the derivation of easily computable error bounds. For problems stemming from the discretization of self-adjoint operators with a continuous spectrum, we obtain sharp estimates grounded in potential theory for Pad\'e approximations and justify extrapolation algorithms at no added computational cost. The obtained results are illustrated on large-scale examples of 2D diffusion and 3D Maxwell's equations as well as a graph from the SNAP repository. We also present promising experimental results on convergence acceleration via random enrichment of the initial block $B$.

翻译：本文探讨了用于计算 $B^T \phi(A) B$ 的求积法，其中 $A$ 是 $\mathbb{R}^{n \times n}$ 中的对称非负定矩阵，$B$ 是 $\mathbb{R}^{n \times p}$ 中的高矩阵，$\phi(\cdot)$ 表示在 $A$ 的谱邻域内足够正则的矩阵函数，例如 Stieltjes 函数或指数函数。这类表达式常见于扩散偏微分方程的多输入多输出（MIMO）传递函数的计算中。我们提出了一种利用块 Lanczos 算法及其通过 Stieltjes 矩阵连分式的等价表示来近似 $B^T \phi(A) B$ 的方案。我们将 Gauss-Radau 求积法的概念推广到块情形，从而便于推导易于计算的误差界。对于源于具有连续谱的自伴算子离散化的问题，我们基于 Padé 近似的位势理论获得了尖锐的估计，并证明了在无需额外计算成本的情况下进行外推算法的合理性。所得结果通过二维扩散方程、三维麦克斯韦方程组的大规模算例以及来自 SNAP 存储库的一个图进行了说明。我们还展示了通过初始块 $B$ 的随机富集来加速收敛的具有前景的实验结果。