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: ∫ − 1 + 1 f ( x ) 1 − x 2 d x {\displaystyle \int _{-1}^{+1}{\frac {f(x)}{\sqrt {1-x^{2}}}}\,dx} and ∫ − 1 + 1 1 − x 2 g ( x ) d x . {\displaystyle \int _{-1}^{+1}{\sqrt {1-x^{2}}}g(x)\,dx.} In the first case ∫ − 1 + 1 f ( x ) 1 − x 2 d x ≈ ∑ i = 1 n w i f ( x i ) {\displaystyle \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 ⁡ ( 2 i − 1 2 n π ) {\displaystyle x_{i}=\cos \left({\frac {2i-1}{2n}}\pi \right)} and the weight w i = π n .

Source: Wikipedia — Chebyshev–Gauss quadrature (CC BY-SA 4.0)

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: ∫ − 1 + 1 f ( x ) 1 − x 2 d x {\displaystyle \int _{-1}^{+1}{\frac {f(x)}{\sqrt {1-x^{2}}}}\,dx} and ∫ − 1 + 1 1 − x 2 g ( x ) d x . {\displaystyle \int _{-1}^{+1}{\sqrt {1-x^{2}}}g(x)\,dx.} In the first case ∫ − 1 + 1 f ( x ) 1 − x 2 d x ≈ ∑ i = 1 n w i f ( x i ) {\displaystyle \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 ⁡ ( 2 i − 1 2 n π ) {\displaystyle x_{i}=\cos \left({\frac {2i-1}{2n}}\pi \right)} and the weight w i = π n .

Source: Wikipedia "Chebyshev–Gauss quadrature" · CC BY-SA 4.0

Share this article: X · Bluesky
Privacy Policy