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:

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

and

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

In the first case

$\underset{-1}{\overset{+1}{\int }}\frac{f(x)}{\sqrt{1-x^2}}dx\approx \sum _{i=1}^n{w}_{i}f(x_i)$

where

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

and the weight

$w_i=\frac{\pi }{n}.$

In the second case

$\underset{-1}{\overset{+1}{\int }}\sqrt{1-x^2}g(x)dx\approx \sum _{i=1}^n{w}_{i}g(x_i)$

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.