The need to evaluate expressions of the form $\mathcal{I}\left(f\right)≔$ trace $\left(W^{T}f\left(A\right)W\right)$, where the matrix $A\in {\mathbb{R}}^{n\times n}$ is symmetric, $W\in {\mathbb{R}}^{n\times k}$ with $1\le k\ll n$, and $f$ is a function defined on the convex hull of the spectrum of $A$, arises in many applications including network analysis and machine learning. When the matrix $A$ is large, the evaluation of $\mathcal{I}\left(f\right)$ by first computing $f\left(A\right)$ may be prohibitively expensive. In this situation it is attractive to compute an approximation of $\mathcal{I}\left(f\right)$ by first applying a few steps of a global Lanczos-type method to reduce $A$ to a small matrix and then evaluating $f$ at this reduced matrix. The computed approximation can be interpreted as a quadrature rule. The present paper generalizes the extended global Lanczos method introduced in Bentbib et al. (2018) and discusses the computation of error-bounds and error estimates. Numerical examples illustrate the performance of the techniques described.

需要评估表格的表达式 $\mathcal{一世}（F）≔$ 跟踪 $（{\mathrm{w\; ^}}^{Ť}F（\mathrm{一种}）\mathrm{w\; ^}）$，其中矩阵 $\mathrm{一种}\in {\mathbb{[R}}^{ñ\times ñ}$ 是对称的 $\mathrm{w\; ^}\in {\mathbb{[R}}^{ñ\times ķ}$ 与 $\mathrm{1个}\le ķ\ll ñ$和 $F$ 是定义在频谱的凸包上的函数 $\mathrm{一种}$出现在许多应用程序中，包括网络分析和机器学习。当矩阵$\mathrm{一种}$ 很大，对 $\mathcal{一世}（F）$ 通过首先计算 $F（\mathrm{一种}）$可能太贵了。在这种情况下，计算$\mathcal{一世}（F）$ 首先应用全局Lanczos型方法的一些步骤来减少 $\mathrm{一种}$ 到一个小的矩阵，然后求值 $F$在这个简化的矩阵上。计算出的近似值可以解释为正交规则。本文概括了Bentbib等人引入的扩展全局Lanczos方法。（2018），并讨论了误差范围和误差估计的计算。数值示例说明了所描述技术的性能。