On the construction of Gaussian quadrature rules from modified moments.

Abstract:

Given a weight function $\omega (x)$ on $(\alpha ,\beta )$, and a system of polynomials $\left \{ {{p_k}(x)} \right \}_{k = 0}^\infty$, with degree ${p_k}(x) = k$, we consider the problem of constructing Gaussian quadrature rules $\int _\alpha ^\beta {f(x)\omega (x)dx = \sum \nolimits _{r = 1}^n {{\lambda _r}^{(n)}f({\xi _r}^{(n)})} }$ from "modified moments" ${v_k} = \int _\alpha ^\beta {{p_k}(x)\omega (x)dx}$. Classical procedures take ${p_k}(x) = {x^k}$, but suffer from progressive ill-conditioning as $n$ increases. A more recent procedure, due to Sack and Donovan, takes for $\{ {p_k}(x)\}$ 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.