The Golub-Welsch algorithm [ Math. Comp., 23: 221-230 (1969)] has long been assumed symmetric for estimating quadratic forms. Recent research indicates that asymmetric quadrature nodes may be more often and the existence of a practical symmetric quadrature for estimating matrix quadratic form is even doubtful.This paper derives a sufficient condition for symmetric quadrature nodes for estimating quadratic forms involving the Jordan-Wielandt matrices which frequently arise from many applications. The condition is closely related to how to construct an initial vector for the underlying Lanczos process. Applications of such constructive results are demonstrated by estimating the Estrada index in complex network analysis.

翻译：Golub-Welsch算法[Math. Comp., 23: 221-230 (1969)]长期以来被假定为估计二次型的对称求积方法。近期研究表明，非对称求积节点可能更为常见，而用于估计矩阵二次型的实用对称求积方法的存在性甚至值得怀疑。本文推导了估计涉及Jordan-Wielandt矩阵的二次型时对称求积节点的充分条件，这类矩阵广泛出现于众多应用场景。该条件与如何为底层Lanczos过程构造初始向量密切相关。此类构造性结果的应用通过估计复杂网络分析中的Estrada指数得以验证。