Chebyshev–Gauss quadrature

In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind:

${\displaystyle \int _{-1}^{+1}{\frac {f(x)}{\sqrt {1-x^{2}}}}\,dx}$

and

${\displaystyle \int _{-1}^{+1}{\sqrt {1-x^{2}}}g(x)\,dx.}$

In the first case

${\displaystyle \int _{-1}^{+1}{\frac {f(x)}{\sqrt {1-x^{2}}}}\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})}$

where

${\displaystyle x_{i}=\cos \left({\frac {2i-1}{2n}}\pi \right)}$

and the weight

${\displaystyle w_{i}={\frac {\pi }{n}}.}$^[1]

In the second case

${\displaystyle \int _{-1}^{+1}{\sqrt {1-x^{2}}}g(x)\,dx\approx \sum _{i=1}^{n}w_{i}g(x_{i})}$

where

${\displaystyle x_{i}=\cos \left({\frac {i}{n+1}}\pi \right)}$

and the weight

${\displaystyle w_{i}={\frac {\pi }{n+1}}\sin ^{2}\left({\frac {i}{n+1}}\pi \right).\,}$^[2]