On Symmetric Lanczos Quadrature for Trace Estimation

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.