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 . {\displaystyle w_{i}={\frac {\pi }{n}}.} [1]

In the second case

∫ − 1 + 1 1 − x 2 g ( x ) d x ≈ ∑ i = 1 n w i g ( x i ) {\displaystyle \int _{-1}^{+1}{\sqrt {1-x^{2}}}g(x)\,dx\approx \sum _{i=1}^{n}w_{i}g(x_{i})}

where

x i = cos ⁡ ( i n + 1 π ) {\displaystyle x_{i}=\cos \left({\frac {i}{n+1}}\pi \right)}

and the weight

w i = π n + 1 sin 2 ⁡ ( i n + 1 π ) . {\displaystyle w_{i}={\frac {\pi }{n+1}}\sin ^{2}\left({\frac {i}{n+1}}\pi \right).\,} [2]

See also

  • Chebyshev polynomials
  • 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.