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:

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

and

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

In the first case

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

where

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

and the weight

$w_{i}={\frac {\pi }{n}}.$

In the second case

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

where

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

and the weight

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

See also

• Chebyshev polynomials
• Chebyshev nodes

External links

• Chebyshev-Gauss Quadrature from Wolfram MathWorld
• Gauss–Chebyshev type 1 quadrature and Gauss–Chebyshev type 2 quadrature, free software in C++, Fortran, and Matlab.