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:

[math]\displaystyle{ \int_{-1}^{+1} \frac {f(x)} {\sqrt{1 - x^2} }\,dx }[/math]

and

[math]\displaystyle{ \int_{-1}^{+1} \sqrt{1 - x^2} g(x)\,dx. }[/math]

In the first case

[math]\displaystyle{ \int_{-1}^{+1} \frac {f(x)} {\sqrt{1-x^2} }\,dx \approx \sum_{i=1}^n w_i f(x_i) }[/math]

where

[math]\displaystyle{ x_i = \cos \left( \frac {2i-1} {2n} \pi \right) }[/math]

and the weight

[math]\displaystyle{ w_i = \frac {\pi} {n}. }[/math]

In the second case

[math]\displaystyle{ \int_{-1}^{+1} \sqrt{1-x^2} g(x)\,dx \approx \sum_{i=1}^n w_i g(x_i) }[/math]

where

[math]\displaystyle{ x_i = \cos \left( \frac {i} {n+1} \pi \right) }[/math]

and the weight

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

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.