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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

On the construction of Gaussian quadrature rules from modified moments.


Author: Walter Gautschi
Journal: Math. Comp. 24 (1970), 245-260
MSC: Primary 65.55
MathSciNet review: 0285117
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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.


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DOI: https://doi.org/10.1090/S0025-5718-1970-0285117-6
Keywords: Constructive theory of Gaussian quadrature rules, tables of Gaussian quadrature rules, numerical condition, orthogonal polynomials
Article copyright: © Copyright 1970 American Mathematical Society