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]

References

↑ 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.
↑ 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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[AS1-1] 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.

[AS2-2] 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.

Chebyshev–Gauss quadrature

See also

References

External links