Английская Википедия:Chebyshev–Gauss quadrature

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

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

<math>\int_{-1}^{+1} \frac {f(x)} {\sqrt{1 - x^2} }\,dx</math>

and

<math>\int_{-1}^{+1} \sqrt{1 - x^2} g(x)\,dx.</math>

In the first case

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

where

<math>x_i = \cos \left( \frac {2i-1} {2n} \pi \right)</math>

and the weight

<math>w_i = \frac {\pi} {n}.</math>

In the second case

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

where

<math>x_i = \cos \left( \frac {i} {n+1} \pi \right) </math>

and the weight

<math> 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.