The Golub-Welsch algorithm [Math. Comp., 23: 221-230 (1969)] for computing Gaussian quadrature rules is of importance in estimating quadratic forms. Quadrature rules based on this algorithm have long been assumed to be symmetric. Recent research indicates that the presence of asymmetric quadrature nodes may be more often. Such a divergence has led to varying error analyses of the Lanczos quadrature method. Since symmetry often implies simplicity, it is of great interest to ask when do Lanczos iterations generate symmetric quadrature rules. This paper derives a sufficient condition that ensures symmetric quadrature nodes which partially answers the question that when the Ritz values of a symmetric matrix are symmetrically distributed. Additionally, we establish both lower and upper bounds on the disparity between the minimum Lanczos iterations required for symmetric and asymmetric quadrature.

翻译：戈卢布-韦尔施算法[《数学计算》，23: 221-230 (1969)]用于计算高斯求积法则，在估计二次型中具有重要意义。长期以来，基于该算法的求积法则被假定为对称的。近期研究表明，非对称求积节点可能出现得更为频繁。这种分歧导致了对兰乔斯求积方法误差分析的不一致性。由于对称性通常意味着简单性，探究兰乔斯迭代何时生成对称求积法则便具有重要价值。本文推导了一个确保对称求积节点的充分条件，该条件部分回答了对称矩阵的里茨值何时呈对称分布的问题。此外，我们建立了对称与非对称求积所需最小兰乔斯迭代次数差异的下界与上界。