On the construction of Gaussian quadrature rules from modified moments.

Author:
Walter Gautschi

Journal:
Math. Comp. **24** (1970), 245-260

MSC:
Primary 65.55

DOI:
https://doi.org/10.1090/S0025-5718-1970-0285117-6

MathSciNet review:
0285117

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Abstract: Given a weight function on , and a system of polynomials , with degree , we consider the problem of constructing Gaussian quadrature rules from "modified moments" . Classical procedures take , but suffer from progressive ill-conditioning as increases. A more recent procedure, due to Sack and Donovan, takes for a system of (classical) orthogonal polynomials. The problem is then remarkably well-conditioned, at least for finite intervals . 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 . 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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Additional Information

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