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Mathematics of Computation

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Chebyshev type quadrature formulas

Author: David K. Kahaner
Journal: Math. Comp. 24 (1970), 571-574
MSC: Primary 65.55
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.

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Keywords: Numerical quadrature, Chebyshev quadrature, equal-weight quadrature
Article copyright: © Copyright 1970 American Mathematical Society

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