# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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by R. E. Barnhill, J. E. Dennis and G. M. Nielson
Math. Comp. 23 (1969), 437-441 Request permission

## Abstract:

A Chebyshev quadrature is of the form $\int _{ - 1}^1 {w(x)f(x)dx \simeq } c\sum \limits _{k = 1}^n {f({x_k})}$ It is usually desirable that the nodes ${x_k}$ be in the interval of integration and that the quadrature be exact for as many monomials as possible (i.e., the first $n + 1$ monomials). For $n = 1, \cdot \cdot \cdot ,7$ and $9$, such a choice of nodes is possible, but for $n = 8$ and $n > 9$, the nodes are complex. In this note, the idea used is that the ${l^2}$-norm of the deviations of the first $n + 1$ monomials from their moments be a minimum. Numerical calculations are carried out for $n = 8,10$, and $11$ and one interesting feature of the numerical results is that a “multiple” node at the origin is required. The above idea is then generalized to a minimization of the ${l^2}$-norm of the deviations of the first $k$ monomials, $k \geqq n + 1$, including $k = \infty$, and corresponding numerical results are presented.
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