A Chebyshev quadrature is of the form
\[
∫
−
1
1
w
(
x
)
f
(
x
)
d
x
≃
c
∑
k
=
1
n
f
(
x
k
)
\int _{ - 1}^1 {w(x)f(x)dx \simeq } c\sum \limits _{k = 1}^n {f({x_k})}
\]
It is usually desirable that the nodes
x
k
{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
n + 1
monomials). For
n
=
1
,
⋅
⋅
⋅
,
7
n = 1, \cdot \cdot \cdot ,7
and
9
9
, such a choice of nodes is possible, but for
n
=
8
n = 8
and
n
>
9
n > 9
, the nodes are complex. In this note, the idea used is that the
l
2
{l^2}
-norm of the deviations of the first
n
+
1
n + 1
monomials from their moments be a minimum. Numerical calculations are carried out for
n
=
8
,
10
n = 8,10
, and
11
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
{l^2}
-norm of the deviations of the first
k
k
monomials,
k
≧
n
+
1
k \geqq n + 1
, including
k
=
∞
k = \infty
, and corresponding numerical results are presented.