标题： 关于对迹估计的对称Lanczos求积 显示英文标题

标题： On Symmetric Lanczos Quadrature for Trace Estimation

摘要： Golub-Welsch 算法通过从 Lanczos 过程中的对称三对角矩阵生成节点和权重来计算高斯求积规则。 尽管对称 Lanczos 求积（在精确算术下）理论上降低了计算成本，但其在迹估计的实际可行性仍不确定。 本文通过建立 Lanczos 求积对称性的充分必要条件解决了这一不确定性。 对于 Jordan-Wielandt 型矩阵，我们提供了选择 Lanczos 算法初始向量的指导，以确保对称求积节点和权重。 更重要的是，在双部图或有向图的 Estrada 指数计算中，我们的方法不仅可以节省计算成本，还可以确保迹估计器的无偏性。 显示英文摘要

摘要： The Golub-Welsch algorithm computes Gauss quadrature rules with the nodes and weights generated from the symmetric tridiagonal matrix in the Lanczos process. While symmetric Lanczos quadrature (in exact arithmetic) theoretically reduces computational costs, its practical feasibility for trace estimation remains uncertain. This paper resolves this ambiguity by establishing sufficient and necessary conditions for the symmetry of Lanczos quadratrure. For matrices of Jordan-Wielandt type, we provide guidance on selecting initial vectors for the Lanczos algorithm that guarantees symmetric quadrature nodes and weights. More importantly, regarding Estrada index computations in bipartite graphs or directed ones, our method would not only save computational costs, but also ensure the unbiasedness of trace estimators.