Chebyshev–Gauss quadrature

{{Short description|Mathematical mentod}}

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

:$\backslash int\_\{-1\}^\{+1\}\; \backslash frac\; \{f(x)\}\; \{\backslash sqrt\{1\; -\; x^2\}\; \}\backslash ,dx$

and

:$\backslash int\_\{-1\}^\{+1\}\; \backslash sqrt\{1\; -\; x^2\}\; g(x)\backslash ,dx.$

In the first case

:$\backslash int\_\{-1\}^\{+1\}\; \backslash frac\; \{f(x)\}\; \{\backslash sqrt\{1-x^2\}\; \}\backslash ,dx\; \backslash approx\; \backslash sum\_\{i=1\}^n\; w\_i\; f(x\_i)$

where

:$x\_i\; =\; \backslash cos\; \backslash left(\; \backslash frac\; \{2i-1\}\; \{2n\}\; \backslash pi\; \backslash right)$

and the weight

:$w\_i\; =\; \backslash frac\; \{\backslash pi\}\; \{n\}.$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.

In the second case

:$\backslash int\_\{-1\}^\{+1\}\; \backslash sqrt\{1-x^2\}\; g(x)\backslash ,dx\; \backslash approx\; \backslash sum\_\{i=1\}^n\; w\_i\; g(x\_i)$

where

:$x\_i\; =\; \backslash cos\; \backslash left(\; \backslash frac\; \{i\}\; \{n+1\}\; \backslash pi\; \backslash right)$

and the weight

:$w\_i\; =\; \backslash frac\; \{\backslash pi\}\; \{n+1\}\; \backslash sin^2\; \backslash left(\; \backslash frac\; \{i\}\; \{n+1\}\; \backslash pi\; \backslash right).\; \backslash ,$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.