Will Lanczos Iterations Generate Symmetric Quadrature Nodes?

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.