On the Construction of Gaussian Quadrature Rules from Modified Moments

On the Construction of Gaussian Quadrature Rules from Modified Moments

Given a weight function ω(x) on (α, β), and a system of polynomials {p<sub>k</sub>(x)}<sup>∞</sup><sub>k = 0</sub>, with degree p<sub>k</sub>(x) = k, we consider the problem of constructing Gaussian quadrature rules <tex-math>$\int^\beta_\alpha f...

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Veröffentlicht in:Mathematics of Computation. - American Mathematical Society, 1960. - 24(1970), 110, Seite 245-s52
1. Verfasser: Gautschi, Walter (VerfasserIn)
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 1970
Zugriff auf das übergeordnete Werk:Mathematics of Computation
Schlagworte:Constructive theory of Gaussian quadrature rules tables of Gaussian quadrature rules numerical condition orthogonal polynomials Mathematics Physical sciences
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520 |a Given a weight function ω(x) on (α, β), and a system of polynomials {p<sub>k</sub>(x)}<sup>∞</sup><sub>k = 0</sub>, with degree p<sub>k</sub>(x) = k, we consider the problem of constructing Gaussian quadrature rules <tex-math>$\int^\beta_\alpha f(x)\omega(x)dx \doteq = \sum^n_{r = 1} \lambda^{(n)}_r f(\xi^{(n)}_r}$</tex-math> from "modified moments" ν<sub>k</sub> = ∫<sup>β</sup><sub>α</sub> p<sub>k</sub>(x)ω(x)dx. Classical procedures take p<sub>k</sub>(x) = x<sup>k</sup>, but suffer from progressive ill-conditioning as n increases. A more recent procedure, due to Sack and Donovan, takes for {p<sub>k</sub>(x)} a system of (classical) orthogonal polynomials. The problem is then remarkably well-conditioned, at least for finite intervals [ α, β ]. 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. 
540 |a Copyright 1970 American Mathematical Society 
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