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# Gaussian Quadrature Error

## Contents

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

## Gaussian Quadrature Matlab

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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## Gauss Legendre Quadrature Example

Integration> Interactive Entries>Interactive Demonstrations> Gaussian Quadrature Seeks to obtain the best numerical estimate gauss quadrature method numerical integration example of an integral by picking optimal abscissas at which to evaluate the function . The fundamental theorem of Gaussian quadrature gauss hermite quadrature states that the optimal abscissas of the -point Gaussian quadrature formulas are precisely the roots of the orthogonal polynomial for the same interval and weighting function. Gaussian quadrature is optimal because it fits https://en.wikipedia.org/wiki/Gaussian_quadrature all polynomials up to degree exactly. Slightly less optimal fits are obtained from Radau quadrature and Laguerre-Gauss quadrature. interval are roots of1 To determine the weights corresponding to the Gaussian abscissas , compute a Lagrange interpolating polynomial for by letting (1) (where Chandrasekhar 1967 uses instead of ), so (2) Then fitting a Lagrange interpolating polynomial through the points gives (3) for arbitrary points . We http://mathworld.wolfram.com/GaussianQuadrature.html are therefore looking for a set of points and weights such that for a weighting function , (4) (5) with weight (6) The weights are sometimes also called the Christoffel numbers (Chandrasekhar 1967). For orthogonal polynomials with , ..., , (7) (Hildebrand 1956, p.322), where is the coefficient of in , then (8) (9) where (10) Using the relationship (11) (Hildebrand 1956, p.323) gives (12) (Note that Press et al. 1992 omit the factor .) In Gaussian quadrature, the weights are all positive. The error is given by (13) (14) where (Hildebrand 1956, pp.320-321). Other curious identities are (15) and (16) (17) (Hildebrand 1956, p.323). In the notation of Szegö (1975), let be an ordered set of points in , and let , ..., be a set of real numbers. If is an arbitrary function on the closed interval , write the Gaussian quadrature as (18) Here are the abscissas and are the Cotes numbers. SEE ALSO: Chebyshev Quadrature, Chebyshev-Gauss Quadrature, Chebyshev-Radau Quadrature, Fundamental Theorem of Gaussian Quadrature, Hermite-Gauss Quadrature, Jacobi-Gauss Quadrature, Laguerre-Gauss Quadrature, Legendre-Gauss Quadrature, Lobatto Quadrature, Radau Quadrature REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathemati

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