A new type of Chebyshev quadrature

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.