On the construction of Gaussian quadrature rules from modified moments.

Abstract

Given a weight function ω ( x ) \omega (x) on ( α , β ) (\alpha ,\beta ) , and a system of polynomials { p k ( x ) } k = 0 ∞ \left \{ {{p_k}(x)} \right \}_{k = 0}^\infty , with degree p k ( x ) = k {p_k}(x) = k , we consider the problem of constructing Gaussian quadrature rules ∫ α β f ( x ) ω ( x ) d x = ∑ r = 1 n λ r ( n ) f ( ξ r ( n ) ) \int _\alpha ^\beta {f(x)\omega (x)dx = \sum olimits _{r = 1}^n {{\lambda _r}^{(n)}f({\xi _r}^{(n)})} } from "modified moments" v k = ∫ α β p k ( x ) ω ( x ) d x {v_k} = \int _\alpha ^\beta {{p_k}(x)\omega (x)dx} . Classical procedures take p k ( x ) = x k {p_k}(x) = {x^k} , but suffer from progressive ill-conditioning as n n increases. A more recent procedure, due to Sack and Donovan, takes for { p k ( x ) } \{ {p_k}(x)\} a system of (classical) orthogonal polynomials. The problem is then remarkably well-conditioned, at least for finite intervals [ α , β ] [\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 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.