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Chebyshev type quadrature formulas


Author: David K. Kahaner
Journal: Math. Comp. 24 (1970), 571-574
MSC: Primary 65.55
DOI: https://doi.org/10.1090/S0025-5718-1970-0273818-5
MathSciNet review: 0273818
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Abstract: Quadrature formulas of the form

$\displaystyle \int_{ - 1}^1 {f(x)dx \approx \frac{2} {n}\sum\limits_{i = 1}^n {f({x_i}^{(n)})} } $

are associated with the name of Chebyshev. Various constraints may be posed on the formula to determine the nodes $ {x_i}^{(n)}$. Classically the formula is required to integrate $ n$th degree polynomials exactly. For $ n = 8$ and $ n \geqq 10$ this leads to some complex nodes. In this note we point out a simple way of determining the nodes so that the formula is exact for polynomials of degree less than $ n$. For $ n = 8$, $ 10$ and $ 11$ we compare our results with others obtained by minimizing the $ {l^2}$-norm of the deviations of the first $ n + 1$ monomials from their moments and point out an error in one of these latter calculations.

References [Enhancements On Off] (What's this?)

  • [1] R. Barnhill, J. Dennis & G. Nielson, "A new type of Chebyshev quadrature," Math. Comp., v. 23, 1969, p. 437. MR 0242367 (39:3698)
  • [2] A. Meir & A. Sharma, "A variation of the Tchebicheff quadrature problem," Illinois J. Math., v. 11, 1967, pp. 535-546. MR 35 #7058. MR 0216223 (35:7058)
  • [3] D. Kahaner, "Equal weight and almost equal weight quadrature formulas," SIAM J. Numer. Anal., v. 6, 1969, pp. 551-556. MR 0286279 (44:3492)
  • [4] F. B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956, p. 346. MR 17, 788. MR 0075670 (17:788d)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1970-0273818-5
Keywords: Numerical quadrature, Chebyshev quadrature, equal-weight quadrature
Article copyright: © Copyright 1970 American Mathematical Society

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