Chebyshev-Gauss quadrature, also called Chebyshev quadrature, is a Gaussian quadrature over the interval ![[-1,1]](/images/equations/Chebyshev-GaussQuadrature/Inline1.svg)
with weighting function 
(Abramowitz and Stegun 1972, p. 889). The abscissas
for quadrature order 
are given by the roots of the Chebyshev
polynomial of the first kind 
, which occur symmetrically about 0. The weights
are
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the coefficient of 
in 
,
 |
(3)
|
and 
the order-
Lagrange interpolating polynomial
for 
.
For Chebyshev polynomials of the first kind,
 |
(4)
|
so
 |
(5)
|
Additionally,
 |
(6)
|
so
 |
(7)
|
Since
 |
(8)
|
the abscissas are given explicitly by
![x_i=cos[((2i-1)pi)/(2n)].](/images/equations/Chebyshev-GaussQuadrature/NumberedEquation7.svg) |
(9)
|
Since
 |  |  |
(10)
|
 |  |  |
(11)
|
where
 |
(12)
|
all the weights are
 |
(13)
|
The explicit formula is then
![int_(-1)^1(f(x)dx)/(sqrt(1-x^2))=pi/nsum_(k=1)^nf[cos((2k-1)/(2n)pi)]+(2pi)/(2^(2n)(2n)!)f^((2n))(xi).](/images/equations/Chebyshev-GaussQuadrature/NumberedEquation10.svg) |
(14)
|
The following two tables give the numerical and analytic values for the first few points and weights.
 |  |  |
| 2 |  | 1.5708 |
| 3 | 0 | 1.0472 |
 | 1.0472 | |
| 4 |  | 0.785398 |
 | 0.785398 | |
| 5 | 0 | 0.628319 |
 | 0.628319 | |
 | 0.628319 |
| 2 |  |  |
| 3 | 0 |  |
| 3 |  |  |
| 4 |  |  |
| 4 |  |  |
| 5 | 0 |  |
| 5 |  |  |
| 5 |  |  |
