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The blue line is the polynomial y ( x ) = 7 x 3 − 8 x 2 − 3 x + 3 {\displaystyle y(x)=7x^ ω 2-8x^ ω 1-3x+3} , whose integral in [-1, 1] is 2/3. The trapezoidal rule returns the integral of the orange dashed line, equal to y ( gaussian quadrature example − 1 ) + y ( 1 ) = − 10 {\displaystyle y(-1)+y(1)=-10} . The 2-point Gaussian gaussian quadrature formula for numerical integration quadrature rule returns the integral of the black dashed curve, equal to y ( − 1 / 3 ) + y ( 1 / 3 gaussian quadrature pdf ) = 2 / 3 {\displaystyle y({-{\sqrt {\scriptstyle 1/3}}})+y({\sqrt {\scriptstyle 1/3}})=2/3} . Such a result is exact since the green region has the same area as the red regions. In numerical analysis, a quadrature rule is an approximation of the definite integral gaussian quadrature calculator of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi and weights wi for i = 1, ..., n. The domain of integration for such a rule is

conventionally taken as [−1, 1], so the rule is stated as ∫ − 1 1 f ( x ) d x = ∑ i = 1 n w i f ( x i ) . {\displaystyle \int _{-1}^ − 8f(x)\,dx=\sum _ − 7^ − 6w_ − 5f(x_ − 4).} Gaussian quadrature as above will only produce good results if the function f(x) is well approximated by a polynomial function within the range [−1, 1]. The method is not, for example, suitable for functions with singularities. However, if the integrated function can be written as f ( x ) = ω ( x ) g ( x ) {\displaystyle f(x)=\omega (x)g(x)\,} , where g(x) is approximately polynomial and ω(x) is known, then alternative weights w i ′ {\displaystyle w_ ξ 8'} and points x i ′ {\displaystyle x_ ξ 6'} that depend on the weighting function ω(x) may give better results, where ∫ − 1 1 f ( x ) d x = ∫ − 1 1 ω ( x ) g ( x ) d x ≈ ∑ i = 1 n w i ′ g ( x i ′ ) . {\displaystyle \int _{-1}^ ξ 4f(x)\,dx=\int _{-1}^ ξ 3\omega (x)g(x)\,dx\approx \sum _ ξ 2^ ξ 1w_ ξ 0'g(x_ ξ 9').} Common weighting functions include ω ( x ) = 1 / 1 − x 2 {\displaystyle \omega (x)=1/{\sqrt ξ 2}}\,} (Chebyshev–Gauss) and ω ( x ) = e − x 2 {\displaystyle \omega (x)=e^{-x^ ξ 0}} (Gauss–Hermite). It

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