Chebyshev type quadrature formulas

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.