On the Construction of Gaussian Quadrature Rules from Modified Moments

摘要

Given a weight function omega(x) on (alpha, beta), and a system of polynomials (p sub k)(x), k = 0 to infinity, with degree p sub k (x) = k, we consider the problem of constructing Gaussian quadrature rules from 'modified moments'. Classical procedures take p sub k (x) = x, but suffer from progressive ill-conditioning as n increases. A more recent procedure, due to Sack and Donovan, takes for p sub k (x) a system of (classical) orthogonal polynomials. The problem is then remarkably well-conditioned, at least for finite intervals (alpha, beta). In support of this observation, we obtain upper bounds for the respective asymptotic condition number. In special cases, these bounds grow like a fixed power of n. We also derive an algorithm for solving the problem considered which generalizes one due to Golub and Welsch. Finally, some numerical examples are presented.