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

$w_i = \frac {\pi} {n}.$^[1]

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

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

See also

• Chebyshev nodes

References

1. ^ Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions, 10th printing with corrections (1972), Dover, ISBN 978-0-486-61272-0. Equation 25.4.38.
2. ^ Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions, 10th printing with corrections (1972), Dover, ISBN 978-0-486-61272-0. Equation 25.4.40.

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.