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Mathematics of Computation

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



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 \[ \int _{ - 1}^1 {f(x)dx \approx \frac {2} {n}\sum \limits _{i = 1}^n {f({x_i}^{(n)})} } \] 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